On upper semicontinuity of the Allen–Cahn twisted eigenvalues

We give an asymptotic upper bound for the kth twisted eigenvalue of the linearized Allen–Cahn operator in terms of the kth eigenvalue of the Jacobi operator, taken with respect to the minimal surface arising as the asymptotic limit of the zero sets of the Allen–Cahn critical points. We use an argument based on the notion of second inner variation developed in Le (On the second inner variations of Allen–Cahn type energies and applications to local minimizers. J. Math. Pures Appl. (9) 103 (2015) 1317–1345).

Indirect NRDF for Partially Observable Gauss-Markov Processes with MSE Distortion: Complete Characterizations and Optimal Solutions

In this paper we study the problem of characterizing and computing the nonanticipative rate distortion function (NRDF) for partially observable multivariate Gauss-Markov processes with hard mean squared error (MSE) distortion constraints. For the finite time horizon case, we first derive the complete characterization of this problem and its corresponding optimal realization which is shown to be a linear functional of the current time sufficient statistic of the past and current observations signals. We show that when the problem is strictly feasible, it can be computed via semidefinite programming (SDP) algorithm. For time-varying scalar processes with average total MSE distortion we derive an optimal closed form expression by means of a dynamic reverse-waterfilling solution that we also implement via an iterative scheme that convergences linearly in finite time, and a closed-form solution under pointwise MSE distortion constraint. For the infinite time horizon, we give necessary and sufficient conditions to sure that asymptotically the sufficient statistic process of the observation signals achieves a steady-state solution for the corresponding covariance matrices and impose conditions that allow existence of a time-invariant solution. Then, we show that when a finite solution exists in the asymptotic limit, it can be computed via SDP algorithm. We also give strong structural properties on the characterization of the problem in the asymptotic limit that allow for an optimal solution via a reverse-waterfilling algorithm that we implement via an iterative scheme that converges linearly under a finite number of spatial components. Subsequently, we compare the computational time needed to execute for both SDP and reverse-waterfilling algorithms when these solve the same problem to show that the latter is a scalable optimization technique. Our results are corroborated with various simulation studies and are also compared with existing results in the literature.

Asymptotic laws of summands I: square integrable independent random variables

This paper is part of series on self-contained papers in which a large part, if not the full extent, of the asymptotic limit theory of summands of independent random variables is exposed. Each paper of the series may be taken as review exposition but specially as a complete exposition expect a few exterior resources. For graduate students and for researchers (beginners or advanced), any paper of the series should be considered as a basis for constructing new results. The contents are taken from advanced books but the organization and the proofs use more recent tools, are given in more details and do not systematically follow previous one. Sometimes, theorems are completed and innovated

Asymptotic Limit-cycle Analysis of Oscillating Chemical Reactions

The asymptotic limit-cycle analysis of mathematical models for oscillating chemical reactions is presented. In this work, after a brief presentation of mathematical preliminaries applied to the biased Van der Pol oscillator, we consider a two-dimensional model of the Chlorine dioxide Iodine Malonic-Acid (CIMA) reactions and the three-dimensional and two-dimensional Oregonator models of the Belousov-Zhabotinsky (BZ) reactions. Explicit analytical expressions are given for the relaxation-oscillation periods of these chemical reactions that are accurate within 5% of their numerical values. In the two-dimensional CIMA and Oregonator models, we also derive critical parameter values leading to canard explosions and implosions in their associated limit cycles.

A system of local/nonlocal p-Laplacians: The eigenvalue problem and its asymptotic limit as p → ∞

In this work, given p ∈ ( 1 , ∞ ), we prove the existence and simplicity of the first eigenvalue λ p and its corresponding eigenvector ( u p , v p ), for the following local/nonlocal PDE system (0.1) − Δ p u + ( − Δ ) p r u = 2 α α + β λ | u | α − 2 | v | β u in Ω − Δ p v + ( − Δ ) p s v = 2 β α + β λ | u | α | v | β − 2 v in Ω u = 0 on R N ∖ Ω v = 0 on R N ∖ Ω , where Ω ⊂ R N is a bounded open domain, 0 < r , s < 1 and α ( p ) + β ( p ) = p. Moreover, we address the asymptotic limit as p → ∞, proving the explicit geometric characterization of the corresponding first ∞-eigenvalue, namely λ ∞ , and the uniformly convergence of the pair ( u p , v p ) to the ∞-eigenvector ( u ∞ , v ∞ ). Finally, the triple ( u ∞ , v ∞ , λ ∞ ) verifies, in the viscosity sense, a limiting PDE system.

The Metaphoricity of Being and the Question of Sameness: Heidegger and Zwicky

ABSTRACT I draw on Heidegger and Zwicky to challenge the notion that underlying divergent perspectives of an entity there must be something that is ‘the same’; instead, sameness is disclosed within particular world-disclosures. I focus on Heidegger's concepts of world and truth as foundation for thinking about different human worlds. I introduce Zwicky's work on gestalts, internal relations, truth as asymptotic limit, and metaphors; the concept of metaphoricity of Being helps us think through how it is that no thing underlies the different perspectives of a phenomenon, and yet that there are limits for disclosures.