scholarly journals A parametric family of quartic Thue inequalities

2021 ◽  
Vol 27 (4) ◽  
pp. 1-14
Author(s):  
Salah Eddine Rihane ◽  
◽  
Mohand Ouamar Hernane ◽  
Alain Togbé ◽  
◽  
...  
Keyword(s):  
Parametric Family ◽  
Pell Equations ◽  
Thue Equation

Let c\neq 0,-1 be an integer. In this paper, we use the method of Tzanakis to transform the quartic Thue equation x^4 -(c^2+c+4) x^3y +(c^2+c+3) x^2 y^2 +2 xy^3 -y^4 = \mu into systems of Pell equations. Then, we determine all primitive solutions (x,y) with 0<|\mu|\leq |c+1|.

scholarly journals A Finite Mixture Modelling Perspective for Combining Experts’ Opinions with an Application to Quantile-Based Risk Measures

Risks ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 115
Author(s):  
Despoina Makariou ◽  
Pauline Barrieu ◽  
George Tzougas
Keyword(s):  
Decision Making ◽  
Mixture Models ◽  
Finite Mixture Models ◽  
Risk Measures ◽  
Finite Mixture ◽  
Parametric Family ◽  
Collective Decision Making ◽  
Multiple Sources ◽  
Mixture Modelling ◽  
Expert Opinions

The key purpose of this paper is to present an alternative viewpoint for combining expert opinions based on finite mixture models. Moreover, we consider that the components of the mixture are not necessarily assumed to be from the same parametric family. This approach can enable the agent to make informed decisions about the uncertain quantity of interest in a flexible manner that accounts for multiple sources of heterogeneity involved in the opinions expressed by the experts in terms of the parametric family, the parameters of each component density, and also the mixing weights. Finally, the proposed models are employed for numerically computing quantile-based risk measures in a collective decision-making context.

A parametric family of Massey-type methods: inference, prediction, and sensitivity

2020 ◽  
Vol 16 (3) ◽  
pp. 255-269
Author(s):  
Enrico Bozzo ◽  
Paolo Vidoni ◽  
Massimo Franceschet
Keyword(s):  
Quantitative Analysis ◽  
Control Theory ◽  
Dynamic Systems ◽  
Parametric Family ◽  
Time Varying ◽  
Multi Agent ◽  
Key Features ◽  
The Stability ◽  
Agent Coordination ◽  
Time Aware

AbstractWe study the stability of a time-aware version of the popular Massey method, previously introduced by Franceschet, M., E. Bozzo, and P. Vidoni. 2017. “The Temporalized Massey’s Method.” Journal of Quantitative Analysis in Sports 13: 37–48, for rating teams in sport competitions. To this end, we embed the temporal Massey method in the theory of time-varying averaging algorithms, which are dynamic systems mainly used in control theory for multi-agent coordination. We also introduce a parametric family of Massey-type methods and show that the original and time-aware Massey versions are, in some sense, particular instances of it. Finally, we discuss the key features of this general family of rating procedures, focusing on inferential and predictive issues and on sensitivity to upsets and modifications of the schedule.

scholarly journals SUSUSY Quantum Mechanics

1997 ◽  
Vol 12 (01) ◽  
pp. 171-176 ◽  
Author(s):  
David J. Fernández C.
Keyword(s):  
Quantum Mechanics ◽  
Shift Operator ◽  
Second Order ◽  
Parametric Family ◽  
First Order ◽  
Shift Operators ◽  
Exactly Solvable ◽  
Exactly Solvable Hamiltonians

The exactly solvable eigenproblems in Schrödinger quantum mechanics typically involve the differential "shift operators". In the standard supersymmetric (SUSY) case, the shift operator turns out to be of first order. In this work, I discuss a technique to generate exactly solvable eigenproblems by using second order shift operators. The links between this method and SUSY are analysed. As an example, we show the existence of a two-parametric family of exactly solvable Hamiltonians, which contains the Abraham–Moses potentials as a particular case.

scholarly journals A parametric family of cubic Thue equations

2004 ◽  
Vol 107 (1) ◽  
pp. 63-79 ◽  
Author(s):  
Alain Togbé
Keyword(s):  
Parametric Family

A parametric family of two ranked objects auctions: equilibria and associated risk

2013 ◽  
Vol 225 (1) ◽  
pp. 141-160 ◽  
Author(s):  
Estrella Alonso ◽  
Joaquin Sanchez-Soriano ◽  
Juan Tejada
Keyword(s):  
Parametric Family

Power Integral Bases in Composits of Number Fields

1998 ◽  
Vol 41 (2) ◽  
pp. 158-165 ◽  
Author(s):  
István Gaál
Keyword(s):  
Number Fields ◽  
Parametric Family ◽  
Prime Degree ◽  
Index Form ◽  
Form Equation ◽  
Integral Bases ◽  
Quadratic Extensions ◽  
Totally Real

AbstractIn the present paper we consider the problem of finding power integral bases in number fields which are composits of two subfields with coprime discriminants. Especially, we consider imaginary quadratic extensions of totally real cyclic number fields of prime degree. As an example we solve the index form equation completely in a two parametric family of fields of degree 10 of this type.

scholarly journals Solution of certain Pell equations

2018 ◽  
Vol 11 (04) ◽  
pp. 1850056 ◽  
Author(s):  
Zahid Raza ◽  
Hafsa Masood Malik
Keyword(s):  
Positive Integer ◽  
Fundamental Solution ◽  
Positive Solutions ◽  
Continued Fraction ◽  
Pell Equation ◽  
Lucas Sequences ◽  
Pell Equations ◽  
Integer Solutions ◽  
Positive Integers

Let [Formula: see text] be any positive integers such that [Formula: see text] and [Formula: see text] is a square free positive integer of the form [Formula: see text] where [Formula: see text] and [Formula: see text] The main focus of this paper is to find the fundamental solution of the equation [Formula: see text] with the help of the continued fraction of [Formula: see text] We also obtain all the positive solutions of the equations [Formula: see text] and [Formula: see text] by means of the Fibonacci and Lucas sequences.Furthermore, in this work, we derive some algebraic relations on the Pell form [Formula: see text] including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation [Formula: see text] in terms of [Formula: see text] We extend all the results of the papers [3, 10, 27, 37].

The Estimation of Nonparametric Functions in a Hilbert Space

1985 ◽  
Vol 1 (1) ◽  
pp. 7-26 ◽  
Author(s):  
A. R. Bergstrom
Keyword(s):  
Hilbert Space ◽  
Nonlinear Regression ◽  
Theoretical Basis ◽  
Fourier Coefficients ◽  
Orthogonal Series ◽  
Regression Function ◽  
Stopping Rule ◽  
Parametric Family ◽  
Convergence Theorems

This paper is concerned with the estimation of a nonlinear regression function which is not assumed to belong to a prespecified parametric family of functions. An orthogonal series estimator is proposed, and Hilbert space methods are used in the derivation of its properties and the proof of several convergence theorems. One of the main objectives of the paper is to provide the theoretical basis for a practical stopping rule which can be used for determining the number of Fourier coefficients to be estimated from a given sample.

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