scholarly journals Cluster algebras and continued fractions

2017 ◽  
Vol 154 (3) ◽  
pp. 565-593 ◽  
Author(s):  
İlke Çanakçı ◽  
Ralf Schiffler
Keyword(s):  
Asymptotic Behavior ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Cluster Algebra ◽  
Perfect Matchings ◽  
Cluster Algebras ◽  
Laurent Polynomials ◽  
Periodic Continued Fractions

We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To a continued fraction $[a_{1},a_{2},\ldots ,a_{n}]$ we associate a snake graph ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ such that the continued fraction is the quotient of the number of perfect matchings of ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ and ${\mathcal{G}}[a_{2},\ldots ,a_{n}]$. We also show that snake graphs are in bijection with continued fractions. We then apply this connection between cluster algebras and continued fractions in two directions. First we use results from snake graph calculus to obtain new identities for the continuants of continued fractions. Then we apply the machinery of continued fractions to cluster algebras and obtain explicit direct formulas for quotients of elements of the cluster algebra as continued fractions of Laurent polynomials in the initial variables. Building on this formula, and using classical methods for infinite periodic continued fractions, we also study the asymptotic behavior of quotients of elements of the cluster algebra.

scholarly journals New Combinatorial Formulas for Cluster Monomials of Type $A$ Quivers

10.37236/6464 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Kyungyong Lee ◽  
Li Li ◽  
Ba Nguyen
Keyword(s):  
Cluster Algebra ◽  
Theta Functions ◽  
Type A ◽  
Perfect Matchings ◽  
Cluster Algebras ◽  
Combinatorial Formula ◽  
Natural Basis ◽  
Combinatorial Formulas ◽  
New Formula

Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type $A$ cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial formula for the cluster monomials in terms of the so-called globally compatible collections. We give bijective proofs of these formulas by comparing with the well-known combinatorial models of the $T$-paths and of the perfect matchings in a snake diagram. For cluster variables of a type $A$ cluster algebra, we give a bijection that relates our new formula with the theta functions constructed by Gross, Hacking, Keel and Kontsevich.

scholarly journals Application of Wynn's epsilon algorithm to periodic continued fractions

1987 ◽  
Vol 30 (2) ◽  
pp. 295-299 ◽  
Author(s):  
M. J. Jamieson
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Quadratic Surd ◽  
Partial Quotients ◽  
Infinite Continued Fraction ◽  
Image Position ◽  
Epsilon Algorithm ◽  
Periodic Continued Fractions

The infinite continued fractionin whichis periodic with period l and is equal to a quadratic surd if and only if the partial quotients, ak, are integers or rational numbers [1]. We shall also assume that they are positive. The transformation discussed below applies only to pure periodic fractions where n is zero.

Linearly periodic continued fractions

2021 ◽  
Vol 105 (564) ◽  
pp. 442-449
Author(s):  
Kantaphon Kuhapatanakul ◽  
Lalitphat Sukruan
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Positive Integers ◽  
Periodic Continued Fractions

An infinite simple continued fraction representation of a real number α is in the form $$\eqalign{& {a_0} + {1 \over {{a_1} + {1 \over {{a_2} + {1 \over {{a_3} + {1 \over {}}}}}}}} \cr & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ddots \cr} $$ where $${a_0}$$ is an integer, and $${a_i}$$ are positive integers for $$i \ge 1$$. This is often written more compactly in one of the following ways: $${a_0} + {1 \over {{a_1} + }}{1 \over {{a_2} + }}{1 \over {{a_3} + }} \ldots \;{\rm{or}}\;\left[ {{a_0};\;{a_1},\;{a_2},\;{a_3} \ldots } \right]$$ .

scholarly journals A Note on Continued Fractions

1960 ◽  
Vol 12 ◽  
pp. 303-308 ◽  
Author(s):  
A. Oppenheim
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Quadratic Irrational ◽  
Image Position ◽  
Positive Integers ◽  
Periodic Continued Fractions

Any real number y leads to a continued fraction of the type(1)where ai, bi are integers which satisfy the inequalities(2)by means of the algorithm(3)the a's being assigned positive integers. The process terminates for rational y; the last denominator bk satisfying bk ≥ ak + 1. For irrational y, the process does not terminate. For a preassigned set of numerators ai ≥ 1, this C.F. development of y is unique; its value being y.Bankier and Leighton (1) call such fractions (1), which satisfy (2), proper continued fractions. Among other questions, they studied the problem of expanding quadratic surds in periodic continued fractions. They state that “it is well-known that not only does every periodic regular continued fraction represent a quadratic irrational, but the regular continued fraction expansion of a quadratic irrational is periodic.

Irrationality measures for almost periodic continued fractions

2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Silvia Dodulíková ◽  
Jaroslav Hančl ◽  
Ondřej Kolouch ◽  
Marko Leinonen ◽  
Kalle Leppälä
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Almost Periodic ◽  
Continued Fraction Expansions ◽  
Periodic Continued Fractions

AbstractWe give general results which provide asymptotic irrationality measures and estimations for the denominators of the convergents for certain almost periodic simple continued fraction expansions. As an application we obtain new irrationality measures, for example, to Napier's constant

CONTINUED FRACTIONS AND JACOBI SYMBOLS

2011 ◽  
Vol 07 (06) ◽  
pp. 1543-1555 ◽  
Author(s):  
KURT GIRSTMAIR
Keyword(s):  
Rational Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Jacobi Symbol ◽  
Right Hand ◽  
Periodic Continued Fractions

We consider the regular continued fraction expansion of a rational number m/N, m ≥ 0, N ≥ 1, (m, N) = 1. Let s/t, (s, t) = 1, be the kth convergent of this expansion and p/q, (p, q) = 1, be the complete quotient belonging to s/t. We give some relations for Jacobi symbols, a typical example of which is [Formula: see text] for k, t, q, N odd, with a simple right-hand side depending on t, q, N (mod 4). As an application, we prove the periodicity of the Jacobi symbol [Formula: see text] for the convergents s/t of infinite purely periodic continued fractions.

scholarly journals Non-periodic continued fractions for quadratic irrationalities

2016 ◽  
Vol 12 (05) ◽  
pp. 1329-1344
Author(s):  
Michael O. Oyengo
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Geometric Series ◽  
Quadratic Irrational ◽  
Number Formula ◽  
Periodic Continued Fractions

A well-known theorem of Lagrange states that the simple continued fraction of a real number [Formula: see text] is periodic if and only if [Formula: see text] is a quadratic irrational. We examine non-periodic and non-simple continued fractions formed by two interlacing geometric series and show that in certain cases they converge to quadratic irrationalities. This phenomenon is connected with certain sequences of polynomials whose properties we examine further.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

scholarly journals Extreme Value Theory for Hurwitz Complex Continued Fractions

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 840
Author(s):  
Maxim Sølund Kirsebom
Keyword(s):  
Invariant Measure ◽  
Extreme Value Theory ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Value Theory ◽  
Extreme Value ◽  
Poisson Law ◽  
Complex Continued Fractions

The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper, we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we obtained several results concerning the extremes of nearest integer continued fractions as well.

scholarly journals Solution of tetrahedron equation and cluster algebras

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
P. Gavrylenko ◽  
M. Semenyakin ◽  
Y. Zenkevich
Keyword(s):  
Integrable System ◽  
Cluster Algebra ◽  
Newton Polygon ◽  
Cluster Algebras ◽  
Weyl Groups ◽  
Newton Polygons ◽  
Tetrahedron Equation ◽  
Affine Weyl Groups ◽  
New Formalism ◽  
Bruhat Cell

Abstract We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

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