Some Classes of Logarithmic-Type Imputation Techniques for Handling Missing Data

In this manuscript, three new classes of log-type imputation techniques have been proposed to handle missing data when conducting surveys. The corresponding classes of point estimators have been derived for estimating the population mean. Their properties (Mean Square Errors and bias) have been studied. An extensive simulation study using data generated from normal, Poisson, and Gamma distributions, as well as real dataset, has been conducted to evaluate how the proposed estimator performs in comparison to several contemporary estimators. The results have been summarized, and discussion regarding real-life applications of the estimator follows.

Double-Scale Gevrey Asymptotics for Logarithmic Type Solutions to Singularly Perturbed Linear Initial Value Problems

We examine a family of linear partial differential equations both singularly perturbed in a complex parameter and singular in complex time at the origin. These equations entail forcing terms which combine polynomial and logarithmic type functions in time and that are bounded holomorphic on horizontal strips in one complex space variable. A set of sectorial holomorphic solutions are built up by means of complete and truncated Laplace transforms w.r.t time and parameter and Fourier inverse integral in space. Asymptotic expansions of these solutions relatively to time and parameter are investigated and two distinguished Gevrey type expansions in monomial and logarithmic scales are exhibited.

Isolated singularities for the n-Liouville equation

In dimension n isolated singularities—at a finite point or at infinity—for solutions of finite total mass to the n-Liouville equation are of logarithmic type. As a consequence, we simplify the classification argument in Esposito (Anal Non Linéaire 35(3):781–801, 2018) and establish a quantization result for entire solutions of the singular n-Liouville equation.

Multiscale Post-Seismic Deformation Based on cGNSS Time Series Following the 2015 Lefkas (W. Greece) Mw6.5 Earthquake

In the present work, a multiscale post-seismic relaxation mechanism, based on the existence of a distribution in relaxation time, is presented. Assuming an Arrhenius dependence of the relaxation time with uniform distributed activation energy in a mesoscopic scale, a generic logarithmic-type relaxation in a macroscopic scale results. The model was applied in the case of the strong 2015 Lefkas Mw6.5 (W. Greece) earthquake, where continuous GNSS (cGNSS) time series were recorded in a station located in the near vicinity of the epicentral area. The application of the present approach to the Lefkas event fits the observed displacements implied by a distribution of relaxation times in the range τmin ≈3.5 days to τmax ≈350 days.

Breather degeneration and lump superposition for the (3 + 1)-dimensional nonlinear evolution equation

This paper is devoted to the study of a (3 + 1)-dimensional generalized nonlinear evolution equation for the shallow-water waves. The breather solutions with different structures are obtained based on the bilinear form with perturbation parameters. Some new lump solitons are found in the process of studying the degradation behavior of breather solutions, and we also study general lump soliton, lumpoff solution and superposition phenomenon between lump soliton and breather solution. Besides, some theorems about the superposition between lump soliton and [Formula: see text]-soliton ([Formula: see text] is a nonnegative integer) are given. Some examples, including lump-[Formula: see text]-exponential type, lump-[Formula: see text]-logarithmic type, higher-order lump-type [Formula: see text]-soliton, are given to illustrate the correctness of the theorems and corollaries described. Finally, some novel nonlinear phenomena, such as emergence of lump soliton, degeneration of breathers, fission and fusion of lumpoff, superposition of lump-[Formula: see text]-solitons, etc., are analyzed and simulated.

Remarks on parabolic De Giorgi classes

We make several remarks concerning properties of functions in parabolic De Giorgi classes of order p. There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex composition, and a logarithmic type estimate. Based on them, we are able to give new proofs of known properties. In particular, we prove local boundedness and local Hölder continuity of these functions via Moser’s ideas, thus avoiding De Giorgi’s heavy machinery. We also seize this opportunity to give a transparent proof of a weak Harnack inequality for nonnegative members of some super-class of De Giorgi, without any covering argument.

A dissipative logarithmic-Laplacian type of plate equation: Asymptotic profile and decay rates

<p style='text-indent:20px;'>We introduce a new model of the logarithmic type of wave like plate equation with a nonlocal logarithmic damping mechanism. We consider the Cauchy problem for this new model in <inline-formula><tex-math id="M1">\begin{document}$ {{\bf R}}^{n} $\end{document}</tex-math></inline-formula>, and study the asymptotic profile and optimal decay rates of solutions as <inline-formula><tex-math id="M2">\begin{document}$ t \to \infty $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M3">\begin{document}$ L^{2} $\end{document}</tex-math></inline-formula>-sense. The operator <inline-formula><tex-math id="M4">\begin{document}$ L $\end{document}</tex-math></inline-formula> considered in this paper was first introduced to dissipate the solutions of the wave equation in the paper studied by Charão-Ikehata [<xref ref-type="bibr" rid="b7">7</xref>]. We will discuss the asymptotic property of the solution as time goes to infinity to our Cauchy problem, and in particular, we classify the property of the solutions into three parts from the viewpoint of regularity of the initial data, that is, diffusion-like, wave-like, and both of them.</p>

Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

<p style='text-indent:20px;'>This work comes to complete some previous ones of ours. Actually, in this paper, we establish some singular weighted inequalities of Trudinger-Moser type for radial functions defined on the whole euclidean space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N,\ N \geq 2. $\end{document}</tex-math></inline-formula> The weights considered are of logarithmic type. The singularity plays a capital role to prove the sharpness of the inequalities. These inequalities are later improved using some concentration-compactness arguments. The last part in this work is devoted to the application of the inequalities established to some singular elliptic nonlinear equations involving a new growth conditions at infinity of exponential type.</p>