scholarly journals Quantum spin probabilities at positive temperature are Hölder Gibbs probabilities

2019 ◽  
Vol 23 (01) ◽  
pp. 1950050
Author(s):  
Jader E. Brasil ◽  
Artur O. Lopes ◽  
Jairo K. Mengue ◽  
Carlos G. Moreira
Keyword(s):  
Continued Fraction ◽  
Quantum Spin ◽  
Positive Temperature ◽  
Stationary Probability ◽  
Zero Temperature ◽  
Continued Fraction Expansion ◽  
Deviation Function ◽  
Spin Lattice ◽  
Shift Map ◽  
Class Of Functions

We consider the KMS state associated to the Hamiltonian [Formula: see text] over the quantum spin lattice [Formula: see text] For a fixed observable of the form [Formula: see text] where [Formula: see text] is self-adjoint, and for positive temperature [Formula: see text] one can get a naturally defined stationary probability [Formula: see text] on the Bernoulli space [Formula: see text]. The Jacobian of [Formula: see text] can be expressed via a certain continued fraction expansion. We will show that this probability is a Gibbs probability for a Hölder potential. Therefore, this probability is mixing for the shift map. For such probability [Formula: see text] we will show the explicit deviation function for a certain class of functions. When decreasing temperature we will be able to exhibit the explicit transition value [Formula: see text] where the set of values of the Jacobian of the Gibbs probability [Formula: see text] changes from being a Cantor set to being an interval. We also present some properties for quantum spin probabilities at zero temperature (for instance, the explicit value of the entropy).

scholarly journals Large deviations for quantum spin probabilities at temperature zero

2018 ◽  
Vol 18 (06) ◽  
pp. 1850044 ◽  
Author(s):  
Artur O. Lopes ◽  
Jairo K. Mengue ◽  
Joana Mohr ◽  
Carlos G. Moreira
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Gibbs State ◽  
Temperature Limit ◽  
Quantum Spin ◽  
Stationary Probability ◽  
Zero Temperature ◽  
Spin Lattice ◽  
Almost Everywhere ◽  
Class Of Functions

We consider certain self-adjoint observable for the KMS state associated to the Hamiltonian [Formula: see text] over the quantum spin lattice [Formula: see text]. For a fixed observable of the form [Formula: see text], where [Formula: see text], and for the zero temperature limit one can get a naturally defined stationary probability [Formula: see text] on the Bernoulli space [Formula: see text]. This probability is ergodic but it is not mixing for the shift map. It is not a Gibbs state for a continuous normalized potential but its Jacobian assume only two values almost everywhere. Anyway, for such probability [Formula: see text] we can show that a Large Deviation Principle is true for a certain class of functions. The result is derived by showing the explicit form of the free energy which is differentiable.

The Solvability of Polynomial Pell’s Equation

2020 ◽  
Vol 25 (2) ◽  
pp. 125-132
Author(s):  
Bal Bahadur Tamang ◽  
Ajay Singh
Keyword(s):  
Trivial Solution ◽  
Continued Fraction ◽  
Laurent Series ◽  
Quadratic Polynomial ◽  
Continued Fraction Expansion ◽  
Pell’S Equation ◽  
Integer Polynomials

This article attempts to describe the continued fraction expansion of ÖD viewed as a Laurent series x-1. As the behavior of the continued fraction expansion of ÖD is related to the solvability of the polynomial Pell’s equation p2-Dq2=1  where D=f2+2g  is monic quadratic polynomial with deg g<deg f  and the solutions p, q  must be integer polynomials. It gives a non-trivial solution if and only if the continued fraction expansion of ÖD  is periodic.

scholarly journals Quantum Spin Systems at Positive Temperature

2006 ◽  
Vol 269 (3) ◽  
pp. 611-657 ◽  
Author(s):  
Marek Biskup ◽  
Lincoln Chayes ◽  
Shannon Starr
Keyword(s):  
Quantum Spin ◽  
Spin Systems ◽  
Positive Temperature ◽  
Quantum Spin Systems

scholarly journals Comparative Study of Discrete Component Realizations of Fractional-Order Capacitor and Inductor Active Emulators

2018 ◽  
Vol 27 (11) ◽  
pp. 1850170 ◽  
Author(s):  
Georgia Tsirimokou ◽  
Aslihan Kartci ◽  
Jaroslav Koton ◽  
Norbert Herencsar ◽  
Costas Psychalinos
Keyword(s):  
Comparative Study ◽  
Fractional Order ◽  
High Performance ◽  
Continued Fraction ◽  
Transfer Functions ◽  
Operational Amplifiers ◽  
Current Conveyors ◽  
Continued Fraction Expansion ◽  
Current Feedback ◽  
Discrete Component

Due to the absence of commercially available fractional-order capacitors and inductors, their implementation can be performed using fractional-order differentiators and integrators, respectively, combined with a voltage-to-current conversion stage. The transfer function of fractional-order differentiators and integrators can be approximated through the utilization of appropriate integer-order transfer functions. In order to achieve that, the Continued Fraction Expansion as well as the Oustaloup’s approximations can be utilized. The accuracy, in terms of magnitude and phase response, of transfer functions of differentiators/integrators derived through the employment of the aforementioned approximations, is very important factor for achieving high performance approximation of the fractional-order elements. A comparative study of the accuracy offered by the Continued Fraction Expansion and the Oustaloup’s approximation is performed in this paper. As a next step, the corresponding implementations of the emulators of the fractional-order elements, derived using fundamental active cells such as operational amplifiers, operational transconductance amplifiers, current conveyors, and current feedback operational amplifiers realized in commercially available discrete-component IC form, are compared in terms of the most important performance characteristics. The most suitable of them are further compared using the OrCAD PSpice software.

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