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Purpose This paper aims to identify the most common fungal species that grow on paper manuscripts and cause bio-deterioration. It also detects the impact of additive materials on fungal degrading and builds a wide database. Thus, it helps conservators understand this phenomenon. Design/methodology/approach In total, 15 samples were collected from different paper manuscripts suffering from fungi. The achievement of the study objectives depends on using a variety of analyzes, such as a microbiological study, which indicated that three main fungi grew on the paper samples. Then, a digital microscope and an environmental scanning electron microscope were used to detect the effect of fungi on paper fibers. Fourier transform infrared microscopy was used to identify the binding medium and the X-ray diffraction method was used to measure the crystallinity index of cellulose of the paper samples. Findings Arabic gum was used as a binder medium with the samples. Aspergillus Niger, Aspergillus Fumigatus and Aspergillus Clavatus were the most common fungal species that grew on the Qur’an papers under investigation. They also caused much common damage to the paper samples. The results of the analyzes also showed that the highest crystallinity index of cellulose was in the samples that contained the lowest rate of fungal growth. Originality/value This paper highlights the relationship between fungal degradation and the multi-component nature of paper manuscripts. It builds a wide database that correlates the composition and the degradation of the Qur’an papers.

Evolutionary game dynamics is a combination of game theory and dynamical systems. Using dynamical theory, we investigate chaotic behavior in asymmetric Rock-Paper-Scissors games under imitative dynamics with two different populations. Our theoretical analysis and numerical simulations demonstrate that the dynamical system can give rise to chaotic behavior in zero-sum and positive-sum asymmetric games. However, chaos cannot occur in a negative-sum asymmetric game. In particular, the numerical simulations show that there is a limit cycle in a negative-sum asymmetric game.

We consider discrete dynamical systems coming from the models of evolution of populations playing rock - paper - scissors game . Asymptotic behaviour of trajectories of these systems is described, occurrence of the Neimark-Sacker bifurcation and nonexistence of time averages are proved.

In a recently published theorem on the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings, Tang et al. (J. Inequal. Appl. 2015:305, 2015) studied a uniformly convex and 2-uniformly smooth real Banach space with the Opial property and best smoothness constant κ satisfying the condition $0 0 < κ < 1 2 , as a real Banach space more general than Hilbert spaces. A well-known example of a uniformly convex and 2-uniformly smooth real Banach space with the Opial property is $E=l_{p}$ E = l p , $2\leq p 2 ≤ p < ∞ . It is shown in this paper that, if κ is the best smoothness constant of E and satisfies the condition $0 0 < κ ≤ 1 2 , then E is necessarily $l_{2}$ l 2 , a real Hilbert space. Furthermore, some important remarks concerning the proof of this theorem are presented.

In this short article, we mainly give some examples to prove that the main results of Section 4 of the paper [W. S. Liao and J. Wu, Ann. Func. Anal. 6 (2015), no. 3, 191–202] are not true. Then we give the corrected matrix inequalities.

This paper analyzes two factoring algorithms, NFS (Number Field Sieve) and ECM (Elliptic Curve Method). The previous results only minimize their running times, however, we may need to minimize the storage size or running time with smaller success probability. We provide these trade-offs, L[s] (\(s\le 1/3\)) memory requires \(L[1-2s]\) running time for NFS, for example. This can be interpreted that NFS requires much more running time when reducing memory complexity.

In this work, we describe a new approach that uses deep neural networks (DNN) to obtain regularization parameters for solving inverse problems. We consider a supervised learning approach, where a network is trained to approximate the mapping from observation data to regularization parameters. Once the network is trained, regularization parameters for newly obtained data can be computed by efficient forward propagation of the DNN. We show that a wide variety of regularization functionals, forward models, and noise models may be considered. The network-obtained regularization parameters can be computed more efficiently and may even lead to more accurate solutions compared to existing regularization parameter selection methods. We emphasize that the key advantage of using DNNs for learning regularization parameters, compared to previous works on learning via optimal experimental design or empirical Bayes risk minimization, is greater generalizability. That is, rather than computing one set of parameters that is optimal with respect to one particular design objective, DNN-computed regularization parameters are tailored to the specific features or properties of the newly observed data. Thus, our approach may better handle cases where the observation is not a close representation of the training set. Furthermore, we avoid the need for expensive and challenging bilevel optimization methods as utilized in other existing training approaches. Numerical results demonstrate the potential of using DNNs to learn regularization parameters., 27 pages, 16 figures

The paper alluded to in the title contains the following striking result: Let $I$ be the unit interval and $\Delta$ the Cantor set. If $X$ is a quasi Banach space containing no copy of $c_0$ which is isomorphic to a closed subspace of a space with a basis and $C(I, X)$ is linearly homeomorphic to $C(\Delta, X)$, then $X$ is locally convex, i.e., a Banach space. It is shown that Kalton result is sharp by exhibiting non locally convex quasi Banach spaces X with a basis for which $C(I, X)$ and $C(\Delta, X)$ are isomorphic. Our examples are rather specific and actually in all cases X is isomorphic to $C(\Delta, X)$ if $K$ is a metric compactum of finite covering dimension., Comment: 4 pages

Oil paper insulation is the main insulation form of large transformers. Partial discharge(PD) in oil paper insulation will not only damage the insulation performance, but also is the precursor and manifestation of insulation deterioration. Weibull distribution parameter is an important characteristic parameter of PD. In this paper a method is proposed to extract the characteristic parameter of PD based on the three-parameter Weibull model, then gray wolf optimization algorithm and particle swarm optimization algorithm are used to calculate the parameters of the model. The result shows that the two algorithms can get the parameters quickly and accurately, and the model can effectively reflect the characteristics of the $H_{n}(q)$ spectrum of PD times.

We discuss the recent paper by A. Dar\'os and L.K. Arruda (On the instability of elliptic traveling wave solutions of the modified Camassa-Holm equation, J. Diff. Equat., 266 (2019), 1946-1968). Our intention is to correct some imperfections left by the authors and present the orbital stability of periodic snoidal waves in the energy space., Comment: 10 pages and 2 figures

In this short review note we show that the new proof of theorem 1.1 given by Zheng Jie Sun and Ling Zhu in the paper Simple proofs of the Cusa-Huygens-type and Becker-Stark-type inequalities is logically incorrect and present another simple proof of the same.

These notes aim to provide a deeper insight on the specifics of two articles dealing with chemotaxis models with nonlinear production. More precisely, we are referring to the papers “Boundedness of solutions to a quasilinear parabolic–parabolic chemotaxis model with nonlinear signal production” by Tao et al. (2019) [2] and “Boundedness for a fully parabolic Keller–Segel model with sublinear segregation and superlinear aggregation” by Frassu and Viglialoro (2021) [1] . These works, independently published in these last years, present results leaving open room for further improvement. Indeed, in the first a gap in the proof of the main claim appears, whereas the cornerstone assumption in the second is not sharp. In these pages we give a more complete picture to the relative underlying comprehension.

Propositions 1.1–1.3 stated below contribute to results and certain problems considered in [E-S], on the behavior of products $$\Pi^n_1(1-z^{a_j}),1\leq{a_1}...\leq{a_n}$$ integers. In the discussion below, {a1,..., an} will be either a proportional subset of {1,..., n} or a set of large arithmetic diameter.

The paper introduces a new taxonomy of ill-posed parametric inverse problems, formulated as global optimization ones. It systematizes irremediable problems, which appear quite often in the real life but cannot be solved using the regularization method. The paper also shows a new way of solving irremediable inverse problems by a complex memetic approach including: genetic computation with adaptive accuracy, random sample clustering and a sophisticated local approximation of misfit plateau regions. Finally, we use a benchmark function featuring cross-shaped plateau to discuss some factors that influence the quality of plateau shape approximation.

We estimate the orientation of wood fibers in porous networks like paper, paperboard or fiberboard by computing digital thermal conductivity experiments on micro-computed tomography ( μ CT) images with artificial isotropic thermal conductivity parameters. The accuracy of mechanical and thermal constitutive models for porous wood fiber based materials crucially depends on knowing the wood fiber orientation. Unfortunately, due to the high porosity, the micro-heterogeneity of wood fibers, the high carbon content of organic materials and the unknown additives present in industrial paper, μ CT-scans often exhibit low contrast and strong artifacts. Conventional image processing approaches encounter difficulties, as they rely upon convex fiber cross sections. We propose a solution by circumventing the segmentation of single wood fibers in μ CT images, by performing thermal conductivity simulations on binarized wood fiber structures, where an artificial isotropic thermal conductivity is associated to the fibers and the pore space is considered as isolating. The local and global temperature fluxes are assembled into a fiber orientation tensor. This method overcomes the limitations of the mentioned local image processing approaches, as individual fibers need not be resolved and convergence for increasing resolution is a consequence of abstract mathematical theory. We use our novel method to analyze large three-dimensional μ CT-scans and a synchrotron scan of a paperboard sample, serving as the starting point of an accurate micromechanical modeling of the effective anisotropic mechanical behavior of paper and paperboard. These results are crucial for calculating the mechanical strength of deep-drawn paperboard, which will be accomplished in a subsequent article.

One of the simplest examples of a robust heteroclinic cycle involves three saddle equilibria: each one is unstable to the next in turn, and connections from one to the next occur within invariant subspaces. Such a situation can be described by a third-order ordinary differential equation (ODE), and typical trajectories approach each equilibrium point in turn, spending progressively longer to cycle around the three points but never stopping. This cycle has been invoked as a model of cyclic competition between populations adopting three strategies, characterised as Rock, Paper and Scissors. When spatial distribution and mobility of the populations is taken into account, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn. The dynamics is described by a set of partial differential equations (PDEs) that has travelling wave (in one dimension) and spiral (in two dimensions) solutions. In this paper, we explore how the robust heteroclinic cycle in the ODE manifests itself in the PDEs. Taking the wavespeed as a parameter, and moving into a travelling frame, the PDEs reduce to a sixth-order set of ODEs, in which travelling waves are created in a Hopf bifurcation and are destroyed in three different heteroclinic bifurcations, depending on parameters, as the travelling wave approaches the heteroclinic cycle. We explore the three different heteroclinic bifurcations, none of which have been observed in the context of robust heteroclinic cycles previously. These results are an important step towards a full understanding of the spiral patterns found in two dimensions, with possible application to travelling waves and spirals in other population dynamics models., Comment: 36 pages, 8 figures

In this paper we consider, discuss and slightly complement recent fixed point results for mappings in b-metric spaces established by Sintunavarat (J Fixed Point Theory Appl 18:397–416, 2016). Thus, all our results are with shorter proofs. In addition, an application to integral equations is given to illustrate the usability of our approach.

This note is devoted to a small, but essential, extension of Theorem 2.1 of our recent paper (LeCrone et al. J Evolut Equ 14:509–533 2014). The improvement is explained in “The improvement” section and proved in “Proof of the main result” section. The importance of the extension is demonstrated in “Application to the Navier–Stokes equations” section with an application to the Navier–Stokes system in critical \(L_q\)-spaces.

Two limit cycles for the continuous model of the rockscissorspaper (RSP) game is constructed.The Hopf bifurcation method is used for the construction of limit cycles.The results give a partial answer to an open question posed by Neumann and Schuster. In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rockscissorspaper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game.

The objective of this contribution is to improve recent stability results for a system coupling ordinary differential equations to a vectorial transport partial differential equation by proposing a new structure of Lyapunov functional. Following the same process of most of the investigations in literature, that are based on an a priori selection of Lyapunov functionals and use the usual integral inequalities (Jensen, Wirtinger, Bessel...), we will present an efficient method to estimate the exponential decay rate of this coupled system leading to a tractable test expressed in terms of linear matrix inequalities. These LMI conditions stem from the new design of a candidate Lyapunov functional, but also the inherent properties of the Legendre polynomials, that are used to build a projection of the infinite dimensional part of the state of the system. Based on these polynomials and using the appropriate Bessel-Legendre inequality, we will prove an exponential stability result and in the end, we will show the efficiency of our approach on academic example.

It is widely recognised that high-stakes assessment can significantly influence what is taught in the classroom. Many argue that high-stakes assessment results in a narrowed curriculum where students learn by rote rather than developing higher cognitive skills. This paper describes a study investigating the various cognitive objectives present from Bloom’s Taxonomy Educational Objectives on the Leaving Certificate biology examination. The study analysed examination papers from the past and current biology syllabuses. Analysis was also carried out to determine the marks being awarded to the different cognitive objectives. The findings show that the examination predominately includes questions that do not promote higher levels of thinking. The majority of the marks on the paper were allocated to the lower objectives of the taxonomy, suggesting students can rely on rote learning to succeed when undertaking the biology examination. This study strongly highlights how high-stake examinations have a narrow scope in terms of student achievement and shows how current biology examination procedures promote low-level learning. This low level of thinking promotes rote learning and regurgitation of facts, requiring little to no understanding of the topics. To prepare students for the working world, there needs to be a shift from only terminal exams to a mixed approach.

This work contains largely afterthoughts, relating to the paper immediately preceding it. We correlate and interpret our contributions in that paper, relative to those of an earlier publication by Aspley, He and McCuan. We propose specific laboratory experiments, suggested by formal predictions of those two papers.

We would like to present the ideas of Makoto Matsumoto and Andras Rapcsak in the projective theory of Finsler spaces. Finally, we give some remarks using of Matsumoto–Rapcsak Theorem.

A criteria due to Toivo Nieminen concerning selfadjoint linear operators in complex Hilbert spaces is extended to the case of linear relations.

We describe a method to construct smooth and compactly supported solutions of 3D incompressible Euler equations and related models. The method is based on localizable Grad–Shafranov equations and is inspired by the recent result (Gavrilov in A steady Euler flow with compact support. Geom Funct Anal 29(1):90–197, [Gav19]).

Some recent analytical papers have explored limiting behaviors of Landau-deGennes models for liquid crystals in certain extreme ranges of the model parameters: limits of "vanishing elasticity" (in the language of some of these papers) and "low-temperature limits." We use simple scaling analysis to show that these limits are properly interpreted as limits in which geometric length scales (such as the size of the domain containing the liquid crystal material) become large compared to intrinsic length scales (such as correlation lengths or coherence lengths, which determine defect core sizes). This represents the natural passage from a mesoscopic model to a macroscopic model and is analogous to a "London limit" in the Ginzburg-Landau theory of superconductivity or a "large-body limit" in the Landau-Lifshitz theory of ferromagnetism. Known relevant length scales in these parameter regimes (nematic correlation length, biaxial coherence length) can be seen to emerge via balances in equilibrium Euler-Lagrange equations associated with well-scaled Landau-deGennes free-energy functionals., 17 pages; no figures; uses Springer journal LaTeX class file svjour3.cls; uses LaTeX packages mathptmx, amsmath, amssymb, and mathtools. Version 'v2' is an update of the note originally posted Dec 2015. The main change (based upon feedback on the earlier note) is to add consideration of the "low temperature" limit (and to illustrate its relationship to the "large body" limit already discussed)

We construct a counterexample that disproves the claim of Theorem 0.2 of the paper “Injective mappings and solvable vector fields of Euclidean spaces”, which appeared in Topology Appl. 136 (2004), 261–274.

In several recent papers, attempts have been made to apply Wardowski's method of $F$-contractions in order to obtain fixed point results for single and multivalued mappings in $b$-metric spaces. In this article, it is shown that in most cases the conditions imposed on respective mappings are too strong and that the results can be obtained directly, i.e., without using most of the properties of auxiliary function $F$.

The aim of this paper is to study a new equivalent form of the weak maximum principle for a large class of differential operators on Riemannian manifolds. This new form has been inspired by the work of Berestycki, Hamel and Rossi for trace operators, and allows us to shed new light on it and to introduce a new sufficient bounded Khas’minskii type condition for its validity. We show its effectiveness by applying it to obtain some uniqueness results in a geometric setting.

Leonetti proved that whenever $${\mathcal {I}}$$ is an ideal on $${\mathbb {N}}$$ such that there exists an uncountable family of sets that are not in $${\mathcal {I}}$$ with the property that the intersection of any two distinct members of that family is in $${\mathcal {I}}$$ , then the space $$c_{0,{\mathcal {I}}}$$ of sequences in $$\ell _\infty $$ that converge to 0 along $${\mathcal {I}}$$ is not complemented. We provide a shorter proof of a more general fact that the quotient space $$\ell _\infty / c_{0,{\mathcal {I}}}$$ does not even embed into $$\ell _\infty $$ .

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

The paper “State-of-the-art of research on seismic pounding between buildings with aligned slabs” (by Kharazian and Lopez-Almansa) has recently been published in the journal of Archives of Computational Methods in Engineering. It can be considered as a kind of continuation of previous ‘state-of-the-art’ publications, including two papers and a book. Unfortunately, the paper contains a number of misleading or wrong statements as well as mistakes. The aim of the present paper is to discuss the controversial statements and to correct the mistakes.

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 teaching a course on verification and validation (V&V) in scientific simulations, it is desirable for students to carry out repeated experiments on devices/systems that they can build and that can be easily repeated. This allows them to be exposed to the inherent aleatory uncertainty associated with building and testing when experiments are used to validate the scientific simulation tools. This paper reports on our experience in using paper helicopters for this purpose in a V&V graduate course. Paper helicopters have been used for various courses, from statistics in high school to graduate optimization courses. They are easily made from paper and paper clips and share the feature of autorotation with real helicopters when they are dropped from altitude. For the V&V course, the helicopters permitted comparison of two models of the drag produced by autorotation that slows their fall. A quadratic dependence of the drag on the speed is generally valid for high Reynolds numbers and a linear model appears for low Reynolds numbers. A gratifying result was that some of the helicopters fitted well the linear model and some fitted better the quadratic model, reflecting the fact that the Reynolds number is in an intermediate range. The paper provides details of how the experiments were conducted and analyzed, which would allow them to be used in similar courses. In addition, actual data are provided, which may be useful for teachers who need to cover the subject in a short time that would not allow the physical experiments. The project also allows a verification component of comparing an analytical solution to one obtained by numerical integration.

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

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $\Theta$ which give rise to self-adjoint Laplacians $-\Delta_{\Theta, \Omega}$ in $L^2(\Omega; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $\Omega\subset\bbR^n$, $n\in\bbN$, $n\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $\Omega$, following an approach introduced by Filonov for this type of problems., Comment: 23 pages, added Remark 5.4

The abstract results and applications presented in “Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable.

Let H be a Hilbert space and let C be a closed convex nonempty subset of H and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T : C\rightarrow H$$\end{document}T:C→H a non-self nonexpansive mapping. A map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h : C\rightarrow R$$\end{document}h:C→R defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(x) := \inf \{\lambda \ge 0 : \lambda x+(1-\lambda )Tx \in C \}$$\end{document}h(x):=inf{λ≥0:λx+(1-λ)Tx∈C}. Then, for a fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in C$$\end{document}x0∈C and for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{\alpha _0} = \max \left\{ {\frac{1}{2},h({x_0})} \right\}\end{aligned}$$\end{document}α0=max12,h(x0), Krasnoselskii–Mann algorithm is defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{n+1}=\alpha _n+(1-\alpha _n)Tx_n,$$\end{document}xn+1=αn+(1-αn)Txn, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{n+1}=\max \{\alpha _n, h(x_{x_{n+1}})\}$$\end{document}αn+1=max{αn,h(xxn+1)}. Recently, Colao and Marino (Fixed Point Theory Appl 2015:39, 2015) have proved both weak and strong convergence theorems when C is a strictly convex set and T is an inward mapping. Meanwhile, they proposed a open question for a countable family of non-self nonexpansive mappings. In this article, authors will give an answer and will prove the further generalized results with the examples to support them.

In this commentary, we try to make clearer the state of the art concerning the relation between mechanical contact interactions and the different notions of stresses. We emphasize the importance of the concept of virtual displacements. Its role has been recognized in Mechanics and in Continuum Mechanics long ago (see e.g., Vailati in Il principio dei lavori virtuali da Aristotele a Erone d’Alessandria, 113–128, 1987; Russo in The forgotten revolution, Springer, Berlin, 2003, or Cosserat and Cosserat in Sur la Theorie des Corps Deformables, Herman, Paris, 1909; Cosserat and Cosserat in Note sur la theorie de l.action euclidienne, Gauthier-Villars, Paris, 1908), and it is central as well when starting with an expression of the power expended by internal stresses and deducing the form of contact interactions as when starting with some form of the contact interactions and developing a representation theorem for these contact interactions based on the Cauchy tetrahedron construction.

A numerical investigation of transient magnetohydrodynamic free convection flow past an infinite vertical plate embedded in a porous medium with viscous dissipation is presented in the above paper. The governing differential equations are transformed into a set of non-linear coupled partial differential equations and are solved numerically using the finite element method. Numerical results for the velocity, temperature and concentration profiles within the boundary layer are presented and discussed.

The purpose of this paper is twofold. Firstly, certain common fixed point theorems are established via $ \phi $-contractive multivalued mappings involving point-dependent control functions as coefficients in the framework of complex valued metric spaces. Our results improve and extend several results in the existing literature. Moreover, this section is equipped by some illustrative examples in support of our results. Secondly, we point out some slip-ups in the examples of some recent papers Ahmad, Klin-Eam, and Azam (2013) and Kutbi, Ahmad, Azam, and Al-Rawashdeh (2014) based on multivalued contractive mappings in complex valued metric spaces. Our observations are also authenticated with the aid of some appropriate examples. Some rectifications to correct the erratic examples are also suggested.

The spectral gap conjecture of M. van den Berg [2, formula (65)] asserts that λ2 − λ1 ≥ 3π for all convex euclidean domains of diameter 1, where λ1 and λ2 denote the first two eigenvalues of the Dirichlet Laplacian. Notice that equality holds for the 1-dimensional unit interval, which can be regarded also as a degenerate n-dimensional rectangular box. The gap estimate is conjectured to hold more generally for Schrodinger operators with convex potentials, under Dirichlet boundary conditions; see the work of S.-T. Yau and collaborators [9, 11]. This Schrodinger gap conjecture was proved some time ago in 1 dimension by R. Lavine [8], and more recently in all dimensions by B. Andrews and J. Clutterbuck [1]. The proof in this journal by H. Donnelly [3] of the original gap conjecture in 2 dimensions (for the Dirichlet Laplacian with zero potential) is not correct. The Editors of Mathematische Zeitschrift have asked us to describe the flaws in the proof, in order to clarify the state of the literature. Donnelly’s approach to the problem is a natural one: first perform a shape optimization to rule out a non-degenerate minimizing domain, and then analyze the spectral gap for a sequence of domains degenerating to an interval, with the help of results by D. Jerison [5]. (For some history on this approach, and on the gap conjecture more generally, see the report on the AIM meeting “Low Eigenvalues of Laplace and Schrodinger Operators” [10], especially page 12 of the open problems list.) The error lies in the proof of the shape optimization step, as we now explain. Donnelly wishes to prove that no minimizing domain can exist for