The undecided have the key: Interaction-driven opinion dynamics in a three state model

The effects of interpersonal interactions on individual's agreements result in a social aggregation process which is reflected in the formation of collective states, as for instance, groups of individuals with a similar opinion about a given issue. This field, which has been a longstanding concern of sociologists and psychologists, has been extended into an area of experimental social psychology, and even has attracted the attention of physicists and mathematicians. In this article, we present a novel model of opinion formation in which agents may either have a strict preference for a choice, or be undecided. The opinion shift emerges during interpersonal communications, as a consequence of a cumulative process of conviction for one of the two extremes opinions through repeated interactions. There are two main ingredients which play key roles in determining the steady state: the initial fraction of undecided agents and the conviction's sensitivity in each interaction. As a function of these two parameters, the model presents a wide range of possible solutions, as for instance, consensus of each opinion, bi-polarisation or convergence of undecided individuals. We found that a minimum fraction of undecided agents is crucial not only for reaching consensus of a given opinion, but also to determine a dominant opinion in a polarised situation. In order to gain a deeper comprehension of the dynamics, we also present the theoretical master equations of the model.Comment: 21 pages, 6 figure

Modeling Opinion Dynamics: Theoretical analysis and continuous approximation

Frequently we revise our first opinions after talking over with other individuals because we get convinced. Argumentation is a verbal and social process aimed at convincing. It includes conversation and persuasion. In this case, the agreement is reached because the new arguments are incorporated. In this paper we deal with a simple model of opinion formation with such persuasion dynamics, and we find the exact analytical solutions for both, long and short range interactions. A novel theoretical approach has been used in order to solve the master equations of the model with non-local kernels. Simulation results demonstrate an excellent agreement with results obtained by the theoretical estimation.Comment: 15 pages, 3 figure

Lower bounds of Fucik eigenvalues of the weighted one dimensional p-Laplacian

In this paper we obtain a family of curves bounding the region which contains all the non trivial Fucik eigenvalues of the weighted one dimensional p laplacian with Neumann boundary conditions. We obtain different proofs of the isolation result of the trivial lines, and the existence of a gap at infinity between the first curve and the trivial lines. We also give a lower bound for the eigenvalues of the p-Laplacian with Neumann boundary conditions

A Note on Price Asymmetry Using a Monetary Model

In this paper we present a macroeconomic foundation of downward money price inflexibility based on classical Monetary Economics. We show that under the principle of risk aversion and the neutral money axiom, our model derives an endogenous asymmetric price response as prices adjust more rapidly when they go upward than downward. This asymmetry does not disappear; on the contrary, it is increasing in time.Fil: Schiaffino, Pablo. Universidad de Palermo; ArgentinaFil: Pinasco, Juan Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentin

Lyapunov-type Inequalities for Partial Differential Equations

In this work we present a Lyapunov inequality for linear and quasilinear elliptic differential operators in $N-$dimensional domains $\Omega$. We also consider singular and degenerate elliptic problems with $A_p$ coefficients involving the $p-$Laplace operator with zero Dirichlet boundary condition. As an application of the inequalities obtained, we derive lower bounds for the first eigenvalue of the $p-$Laplacian, and compare them with the usual ones in the literature

Monty Hall game: a host with limited budget

In this paper we introduce a new version of the classical Monty Hall problem, where the host is trying to maximize the audience while is restricted in its budget. This problem is related to the design of games with a predetermined outcome, and decision making process under uncertainty when the agent does not know if the received advice is favorable or not.Fil: Alvarez, Agustin. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Pinasco, Juan Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentin

Quasilinear eigenvalues

In this work, we review and extend some well known results for the eigenvalues of the Dirichlet $p-$Laplace operator to a more general class of monotone quasilinear elliptic operators. As an application we obtain some homogenization results for nonlinear eigenvalues.Comment: 23 pages, Rev. UMA, to appea

Bounds and asymptotic estimations for the eigenvalues of non linear elliptic problems

[fórmulas aproximadas, revisar las mismas en el original]. En este trabajo obtendremos cotas y estimaciones asintóticas para los autovalores {λk}k del p-Laplaciano unidimensional con una función peso r(t): −(|u’(t)|^p−2 u’(t))’ = λr(t)|u(t)| p−2u(t), con diferentes condiciones de borde. Obtendremos una generalización de la teoría de Sturm Liouville basada en desigualdades integrales, que nos permitirá presentar una demostración de la desigualdad de Lyapunov. Con esta obtendremos cotas inferiores óptimas para autovalores. Daremos otras demostraciones diferentes de esta desigualdad y distintas aplicaciones. Para el problema con condición de borde Neumann y pesos indefinidos demostraremos que los autovalores variacionales son todos. También obtendremos curvas que contienen el espectro de Fučik, y daremos otra demostración de que para esta condición de borde las líneas triviales del espectro son aisladas, y que la segunda curva presenta una separación de los ejes en infinito. Combinando métodos variacionales con la teoría de Sturm Liouville no lineal, obtendremos el desarrollo asintótico de la función N(λ) definida como N(λ) = #{k : λk ≤ λ}. Calcularemos el primer término en el desarrollo de N(λ) y daremos una estimación del segundo término. Aquí, Ω puede ser una unión infinita de intervalos disjuntos, en tal caso, ∂Ω tendrá una dimensión interior de Minkowski d ∈ [0, 1), y el desarrollo será: N(λ) = λ^(1/p)/ 2πp ∫Ω r^(1/p)(t) dt + O(λ^d/p), donde πp = 2(p − 1)^1/p π/p/sin(π/p) . De este desarrollo obtendremos la siguiente fórmula asintótica para el k-ésimo autovalor, λk ∼ (πpk/ ∫Ω r^(1/p)(t) dt)^p. Extenderemos los resultados obtenidos para la función N(λ) para pesos que cambian de signo y a distintos problemas singulares, tales como el comportamiento asintótico de los autovalores radiales en R N de la ecuación −∆pu = −div(|∇u|^p− 2 ∇u) = (λ − q(|x|)|u|^p−2 u, y del problema radial en una bola.[fórmulas aproximadas, revisar las mismas en el original]. This work is concerned with eigenvalue bounds and the asymptotic behaviour of the eigenvalues {λk}k of the weighted p-laplacian equation, −(|u’(t)|^p−2 u’(t))’ = λr(t)|u(t)| p−2u(t), with different boundary conditions (Dirichlet, Neumann, mixed), where Ω ⊂ R is an open set, 1 < p < +∞, λ is the eigenvalue parameter, and r(t) is a real function. We develop a non linear Sturm-Liouville theory with integral inequalities on the weights instead of the classical pointwise conditions. We obtain from it a Lyapunov inequality, which in turns gives optimal lower bounds for the eigenvalues {λk}k. For the Neumann eigenvalue problem with indefinite weights, we prove that the variational eigenvalues exhaust the spectrum. Also, we consider the Fučik spectrum with the Neumann boundary condition, and we will show different proofs of the isolation of the trivial lines and the existence of a gap at infinity. By combining variational methods and the non linear Sturm Liouville theory, we obtain the asymptotic expansion of N(λ), the spectral counting function defined as N(λ) = #{k : λk ≤ λ}. We compute the first term and we give an estimation of the error term. Here, Ω = ∪j∈NIj , and ∂Ω has an associated fractal dimension d ∈ [0, 1). We show that the growth of the error term depends on the interior Minkowski dimension d of ∂Ω: N(λ) = λ^(1/p)/ 2πp ∫Ω r^(1/p)(t) dt + O(λ^d/p), where πp = 2(p − 1)^1/p ∫1 0 ds/(1 − s^p)^1/p . As a corollary, we obtain the asymptotic behavior of eigenvalues: λk ∼ (πpk/ ∫Ω r^(1/p)(t) dt)^p. We extend the previous results to general weights r(t) which are allowed to change signs, and without continuity hypotheses. Also, we consider singular eigenvalue problems related to the asymptotic distribution of radial eigenvalues.Fil: Pinasco, Juan Pablo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina