Juan Pablo Pinasco

Lower bounds for eigenvalues of the one-dimensional

We present sharp lower bounds for eigenvalues of the one-dimensional p-Laplace operator. The method of proof is rather elementary, based on a suitable generalization of the Lyapunov inequality.

p

In this paper we generalize the Sturm comparison theorem for the one dimensional p-Laplacian with integral inequalities instead of pointwise conditions on the coefficients. We apply it to find lower bounds of eigenvalues, and we present a new proof of the Lyapunov inequality.

-Laplacian

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, in a threshold process, as a consequence of a cumulative persuasion for either one of the two opinions in repeated interactions. There are two main ingredients which play key roles in determining the steady states: the initial fraction of undecided agents and the change in agents’ persuasion after each interaction. As a function of these two parameters, the model presents a wide range of solutions, among which there are consensus of each opinion and bi-polarization. We found that a minimum fraction of undecided agents is not crucial for reaching consensus only, but also to determine a dominant opinion in a polarized situation. In order to gain a deeper comprehension of the dynamics, we also present the theoretical framework of the model. The master equations are of special interest for their nontrivial properties and difficulties in being solved analytically.

Comparison of eigenvalues for the p-Laplacian with integral inequalities

Let 𝕋 = {a n } n ∪{0} be a time scale with zero Minkowski (or box) dimension, where {a n } n is a monotonically decreasing sequence converging to zero, and a 1 = 1. In this paper, we find an upper bound for the eigenvalue counting function of the linear problem − u ΔΔ = λu σ, with Dirichlet boundary conditions. We obtain that the nth-eigenvalue is bounded below by . We show that the bound is optimal for the q-difference equations arising in quantum calculus.

The Undecided Have the Key: Interaction-Driven Opinion Dynamics in a Three State Model

We obtain the asymptotic distribution of the nonprincipal eigenvalues associated with the singular problem x″

Detailed asymptotic of eigenvalues on time scales

In this paper we study an inverse problem for weighted second order Sturm-Liouville equations. We show that the zeros of any subsequence of eigenfunctions, or a dense set of nodes, are enough to determine the weight. We impose weaker hypotheses for positive weights, and we generalize the proof to include indefinite weights. Moreover, the parameters in the boundary conditions can be determined numerically by using a shooting method.

The distribution of nonprincipal eigenvalues of singular second-order linear ordinary differential equations

In this note we give a new proof of the existence of infinitely many prime numbers. There are several different proofs with many variants, and some of them can be found in [1, 3, 4, 5, 6]. This proof is based on a simple counting argument using the inclusionexclusion principle combined with an explicit formula. A different proof based on counting arguments is due to Thue (1897) and can be found in [6] together with several generalizations, and a remarkable variant of it was given by Chaitin [2] using algorithmic information theory. Moreover, we prove that the series of reciprocals of the primes diverges. Our proofs arise from a connection between the inclusion-exclusion principle and the infinite product of Euler. Let {pi }i be the sequence of prime numbers, and let us define the following recurrence: a0 = 0, ak+1 = ak + 1 − ak pk+1 .

A nodal inverse problem for second order Sturm-Liouville operators with indefinite weights

We derive oscillation and nonoscillation criteria for the one-dimensional-Laplacian in terms of an eigenvalue inequality for a mixed problem. We generalize the results obtained in the linear case by Nehari and Willett, and the proof is based on a Picone-type identity.

New Proofs of Euclid's and Euler's Theorems

In this work we extend an inequality of Nehari to the eigenvalues of weighted quasilinear problems involving the p-Laplacian when the weight is a monotonic function. We apply it to different eigenvalue problems.

Eigenvalues of the -Laplacian and disconjugacy criteria

In this work, we introduce an energy function in order to study finite scale free graphs generated with different models. The energy distribution has a fractal pattern and presents log periodic oscillations for high energies. These oscillations are related to a discrete scale invariance of certain graphs, that is, there are preferred scaling ratios suggesting a hierarchical distribution of node degrees. On the other hand, small energies correspond to graphs with evenly distributed degrees.