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The aim of this paper is to discuss the main result in the paper by D.Y. Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective function has not local extrema with the mentioned constraints for $p\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer. We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995]., 12 pages; in this version we added the forgotten condition $F(x) \ne 0$ for $x\in (a,b)$ on page 3

Origami is the ancient Japanese art of paper folding. It is possible to fold many intriguing geometrical shapes with paper [M]. In this article, the question we will answer is which shapes are possible to construct and which shapes are impossible to construct using origami. One of the most interesting things we discovered is that it is impossible to construct a cube with twice the volume of a given cube using origami, just as it is impossible to do using a compass and straight edge. As an unexpected surprise, our algebraic characterization of origami is related to David Hilbert's 17th problem. Hilbert's problem is to show that any rational function which is always non-negative is a sum of squares of rational functions [B]. This problem was solved by Artin in 1926 [Ar]. We would like to thank John Tate for noticing the relationship between our present work and Hilbert's 17th problem. This research is the result of a project in the Junior Fellows Program at The University of Texas. The Junior Fellows Program is a program in which a junior undergraduate strives to do original research under the guidance of a faculty mentor. The referee mentioned two references which the reader may find interesting. "Geometric Exercises in Paper Folding" addresses practical problems of paper folding [R]. Arnong many other things, Sundara Row gives constructions for the 5-gon, the 17-gon, and duplicating a cube. His constructions, however, use more general folding techniques than the ones we consider here. Felix Klein cites Row's work in his lectures on selected questions in elementary geometry [K]. In order to understand the rules of origami construction, we will first consider a sheet of everyday notebook paper. Our work with notebook paper will serve as an intuitive model for our definition of origami constructions in the Euclidean plane. There are four natural methods of folding a piece of paper. The methods will serve as the basis of the definition of an origami pair. We construct the line L1, by folding a crease between two different corners of the paper. Another line may be constructed by matching two corners. For example, if corners or and Sy are matched, the crease formed, L2, will be the perpendicular bisector of the segment orSy. Another natural construction is matching one line to another line. For instance, ,8^y, the paper's edge, and L2 are lines. If we lay ,8^y upon L2 and form the crease, then we obtain L3 which is the angle bisector of the two lines. If we start with two parallel lines in this third construction, then we will just get a parallel line half way in between. The fourth and final construction which seems natural is consecutive folding. This is similar to rolling up the sheet of paper only one does not roll it up, he folds it up. More explicitly, start with a piece of paper with two creases on it as in Figure 2. Fold along line Ll and do not unfold the piece of paper. Notice that line L2 lies

In this paper author was proved the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent. As an application we prove the boundedness of certain sublinear operators on the weighted variable Lebesgue space. Note that the proof of multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent don't contained any mistakes. But at the proving of the boundedness of certain sublinear operators on the weighted variable Lebesgue space Georgian colleagues discovered a small but significant error in this paper. This result is assigned as Theorem 5 in noted paper. In other words, sufficient conditions for general weights ensuring the validity of the two-weight strong type inequalities for some sublinear operator was found. In this theorem the inequality (9) isn't true. In this note we give the details of the correct argument. We presume that the reader is familiar with the contents and notation of our original paper. At the heart of our correction is the following Theorem which replaces Theorem 5.

To prove Theorem 2, we use the following two facts about the class F: 1. F is a differential ring [2]. This means that F is closed under addition, multiplication and differentiation. 2. All functions y ∈ F are entire functions of completely regular growth in the sense of Levin–Pfluger [4], with piecewise-trigonometric indicators, the notions which we recall now. Let f be a holomorphic function in an angular sector S = {re : |θ−θ0| 0}. We say that f has completely regular growth with respect to order ρ > 0 if the following finite limit exists

In the paper (HA14), unfortunately, a computational error occurred in one estimate. Although the wrong estimate does not affect the main results, we want to present the necessary corrections. Essentially, Lemma 5.2 has to be corrected and, since it is used in the proof of Theorem 5.1, the proof of this theorem also requires an adaptation. (i) The corrected formulation of Lemma 5.2 is as follows. Lemma 5.2 For Lagrange finite elements with a shape-regular family of affine meshes { T n h } h>0 there is a constant C > 0 independent of q and h such that for all w ∈ Wh and q = 2m, m ∈N: CΛq−2 p (∇w,∇Ip h (wq−1))T ∫ T ‖∇w‖l2‖w‖ q−2 0,∞,T dx, ∀T ∈ T n h , (5.1) where Λp = ‖ ∑ndof i=1 |φi|‖0,∞,T is the Lebesgue constant.

We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freist\"uhler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations, Comment: to appear in Proceedings of the AMS

We show abstract versions for Banach couples of several limiting compact interpolation theorems established by Edmunds and Opic for couples of Lp spaces.

We indicate how recent results of Grafakos and Teschl (J Fourier Anal Appl 19:167–179, 2013), and Estrada (J Fourier Anal Appl 20:301–320, 2014), relating the Fourier transform of a radial function in $$\mathbb R^n$$ and the Fourier transform of the same function in $$\mathbb R^{n+2}$$ and $$\mathbb R^{n+1}$$ , respectively, are located within known results on transplantation for Hankel transforms.

This study aims to investigate plane geometry problem-solving strategies of prospective mathematics teachers using dynamic geometry software (DGS) and paper-and-pencil (PPB) environments after receiving an instruction with GeoGebra (GGB). Four plane geometry problems were used in a multiple case study design to understand the solution strategies developed by 2 prospective teachers. The results revealed that although the participants mostly used algebraic solutions in the PPB environment, they preferred geometric solutions in the GGB environment even though algebraic solutions were still possible (the software did not preclude them). Furthermore, different proofing strategies were developed in each environment. This suggests that changing the environment may prompt students to seek for additional solutions, which, in turn, results in a deeper understanding of the problem. As such, using both environments simultaneously in solving the same problems appears to bring about important benefits.

In this paper, we discuss the validity of the result of Khuangsatung and Kangtunyakarn [Existence and convergence theorem for fixed point problem of various nonlinear mappings and variational inequality problems without some assumptions, Filomat 32(1) (2018) 305–309].

In the paper we give some remarks on the article of Janet Mills. In particular, the proof of Lemma 1.2 (in her work) is incorrect, and so the proof of Theorem 3.5 is not valid, too. Using different methods we show the mentioned theorem. Moreover, we find a new equivalent condition to the statements in Theorem 3.5. In particular, an explicit definition of a new class of orthodox semigroups is introduced.

Motivated by a paper of Fang (2009), we study the Samuel multiplicity and the structure of essentially semi-regular operators on an infinite-dimensional complex Banach space. First, we generalize Fang’s results concerning Samuel multiplicity from semi-Fredholm operators to essentially semi-regular operators by elementary methods in operator theory. Second, we study the structure of essentially semi-regular operators. More precisely, we present a revised version of Fang’s 4 × 4 upper triangular model with a little modification, and prove it in detail after providing numerous preliminary results, some of which are inspired by Fang’s paper. At last, as some applications, we get the structure of semi-Fredholm operators which revised Fang’s 4 × 4 upper triangular model, from a different viewpoint, and characterize a semi-regular point λ ∈ ℂ in an essentially semi-regular domain.

In 1935, Erdos proved that all natural numbers can be written as a sum of a square of a prime and a square-free number. In 1939, Pillai derived an asymptotic formula for the number of such representations. The mathematical review of Pillai’s paper stated that the proof of the above result contained inaccuracies, thus casting a doubt on the correctness of the paper. In this paper, we re-examine Pillai’s paper and show that his argument was essentially correct. Afterwards, we improve the error term in Pillai’s theorem using the Bombieri–Vinogradov theorem.

The proofs of Theorem 2.2 of K. J. Dykema and M. Rørdam, Purely infinite simple C*-algebras arising from free product constructions, Canad. J. Math. 50 (1998), 323–341 and of Theorem 3.1 of K J. Dykema, Purely infinite simple C*-algebras arising from free product constructions, II,Math. Scand. 90 (2002), 73–86 are corrected.