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The logarithmic Schrödinger equation (LogSE) has a logarithmic nonlinearity f(u) = uln | u|² that is not differentiable at u = 0. Compared with its counterpart with a regular nonlinear term, it possesses richer and unusual dynamics, though the low regularity of the nonlinearity brings about significant challenges in both analysis and computation. Among very limited numerical studies, the semi-implicit regularized method via regularizing f(u) as u\varepsilon ln(\varepsilon +| u\varepsilon |)² to overcome the blowup of ln | u| 2 at u = 0 has been investigated recently in the literature. With the understanding of f(0) =0, we analyze the nonregularized first-order implicit-explicit scheme for the LogSE. We introduce some new tools for the error analysis that include the characterization of the H\"older continuity of the logarithmic term, and a nonlinear Gr\"onwall's inequality. We provide ample numerical results to demonstrate the expected convergence. We position this work as the first to study the direct linearized scheme for the LogSE as far as we can tell. [ABSTRACT FROM AUTHOR]

Let f : R→R be continuous. We examine the relationship between the so-called "big Lip" and "little lip" functions: Lip f and lip f. [ABSTRACT FROM AUTHOR]

We provide two new characterizations of the Takagi function as the unique bounded solution of some systems of two functional equations. The results are independent of those obtained by Kairies (Wyż Szkoł Ped Krakow Rocznik Nauk Dydakt Prace Mat 196:73-82, 1998), Kairies (Aequ Math 53:207-241, 1997), Kairies (Aequ Math 58:183-191, 1999) and Kairies et al. (Rad Mat 4:361-374, 1989; Errata, Rad Mat 5:179-180, 1989). [ABSTRACT FROM AUTHOR]

Fractal interpolation functions (FIFs) supplement and subsume all classical interpolants. The major advantage by the use of fractal functions is that they can capture either the irregularity or the smoothness associated with a function. This work proposes the use of cubic spline FIFs through moments for the solutions of a two-point boundary value problem (BVP) involving a complicated non-smooth function in the non-homogeneous second order differential equation. In particular, we have taken a second order linear BVP: y ′ ′ (x) + Q (x) y ′ (x) + P (x) y (x) = R (x) with the Dirichlet's boundary conditions, where P (x) and Q (x) are smooth, but R (x) may be a continuous nowhere differentiable function. Using the discretized version of the differential equation, the moments are computed through a tridiagonal system obtained from the continuity conditions at the internal grids and endpoint conditions by the derivative function. These moments are then used to construct the cubic fractal spline solution of the BVP, where the non-smooth nature of y ′ ′ can be captured by fractal methodology. When the scaling factors associated with the fractal spline are taken as zero, the fractal solution reduces to the classical cubic spline solution of the BVP. We prove that the proposed method is convergent based on its truncation error analysis at grid points. Numerical examples are given to support the advantage of the fractal methodology. [ABSTRACT FROM AUTHOR]

Abstract: Classical turnpike theory, as originally conceived by Samuelson, pertains to optimal growth theory over a large but finite time horizon with given initial and terminal stocks. In this paper, we present two turnpike results in the context of the economics of forestry with given initial and terminal forest configurations. Our results depart from the general theory in that they concern a transitional production set which does not satisfy the assumptions of inaction and free disposal, and rely on a recently discovered non-interiority assumption on concave (not necessarily differentiable) benefit functions that implies, and is implied by, the asymptotic convergence of good programs. [Copyright &y& Elsevier]

This paper proposes a fast multilevel method using primal relaxations for the total variation image denoising and analyzes its convergence. The basic primal relaxation is known to get stuck at a nonstationary point (nearly a local minimum) of the minimization, whose solution is known to be "nonsmooth" in the space of functions with bounded variation. Our idea is to use coarse level corrections, overcoming the deadlock in a basic primal relaxation scheme and achieving much improvement over relaxation. Moreover, to reach a global minimizer, further refinement of the multilevel method is needed, and we propose a nonregular coarse level based on a patch-detection idea (relating to hemivariateness) to correct and improve the standard multilevel method. Both algorithmic and analytical results together with numerical experiments on both one- and two-dimensional images are presented. [ABSTRACT FROM AUTHOR]

Abstract: We prove that, for , there are nonnegative functions f of n variables (and even flat ones for ) which are not a finite sum of squares of functions. For , where a decomposition in a sum of two squares is always possible, we investigate the possibility of writing . We prove that, in general, one cannot require a better regularity than . Assuming that f vanishes at all its local minima, we prove that it is possible to get but that one cannot require any additional regularity. [Copyright &y& Elsevier]

The finite difference method and smoothing approximations for a nonsmooth constrained optimal control problem are considered. Convergence of solutions of discretized smoothing optimal control problems is proved. Error estimates of finite difference smoothing solutions are given. Numerical examples are used to test a smoothing sequential quadratic programming (SQP) method for solving the nonsmooth constrained optimal control problem. [ABSTRACT FROM AUTHOR]