On Existence of Common Fixed Points Under Lipschitz-Type Mapping Pairs with Applications

In this paper, we propose some su?cient conditions to obtain the existence of common fixed points for a pair of self-mappings satisfying Lipschitz-type conditions in noncomplete metric space. Our results improve and extend the results of Pant [Common fixed points of Lipschitz-type mapping pairs, J. Math. Anal. Appl. 240, (1999), 280–283] and Khan et al. [Coincidences of Lipschitz-type hybrid maps and invariant approximation, Numer. Funct. Anal. Optim. 28(9–10), (2007), 1165–1177]. As an application of our results we solve an eigenvalue problem for operators defined on a normed space.     

Asymptotics for eigenvalues of the Laplacian with a Neumann boundary condition on a thin cut-out tubeComportement asymptotique des valeurs propres du laplacien avec conditions de Neumann sur un tube extrait

We consider the eigenvalue problem for the Laplace operator in a bounded three-dimensional domain where a thin tube is cut out. Imposing a Neumann boundary condition on the boundary of this tube, we construct asymptotics for eigenvalues on the small parameter that is a diameter of the tube. To cite this article: M.Yu. Planida, C. R. Mecanique 331 (2003).     

Stability conditions of prestressed pin-jointed structures

Pin-jointed structures are first classified to trusses, tensile structures, and tensegrity structures in view of their respective stability properties. A sufficient condition for stability of an equilibrium state is derived for tensegrity structures. The condition is based on the bilinear forms of the linear and geometrical stiffness matrices considering the flexibility of members. The stability is defined by the positive definiteness of the tangent stiffness matrix, whereas the definition of prestress-stability is based on the geometrical stiffness matrix and the infinitesimal mechanisms. Numerical examples verify that the so-called super-stability condition might not be satisfied by a stable tensegrity structure, and that a prestress-stable structure can be unstable if the prestresses are moderately large.     

Novel matrix invariants for characterization of changes of proteomics maps

Previous studies on mathematical characterization of proteomics maps by sets of map invariants were based on the construction of a set of distance-related matrices obtained by matrix multiplication of a single matrix by itself. Here we consider an alternative characterization of proteomics maps based on a set of matrices characterizing local features of an embedded zigzag curve over the map. It is shown that novel invariants can well characterize proteomics maps. Advantages of the novel approach are discussed.     

Existence,nonexistence and multiplicity results for nonlocal Dirichlet problems

In this paper we establish various existence, nonexistence and multiplicity results for fully nonlinear Dirichlet problems associated to nonlocal Hamilton–Jacobi equations. This study is accomplished by a careful analysis of the principal eigenvalues of the elliptic operator. Resonance phenomena and anti maximum principles are also established.     

Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

We consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold M isometrically immersed into another Riemannian manifold $\overline{M}$. We first assume the pull back Weitzenböck operator of $\overline{M}$ bounded from below, and obtain an extrinsic lower bound for the first eigenvalue of Hodge-Laplacian. As applications, we obtain some rigidity results. Second, when the pull back Weitzenböck operator of $\overline{M}$ bounded from both sides, we give a lower bound of the first eigenvalue by the Ricci curvature of M and some extrinsic geometry. As a consequence, we prove a weak Ejiri type theorem, that is, if the Ricci curvature bounded from below pointwisely by a function of the norm square of the mean curvature vector, then M is a homology sphere. In the end, we give an example to show that all the eigenvalue estimates are optimal when $\overline{M}$ is the space form.     

Elastic stability of all edges simply supported,stepped and stiffened rectangular plate under Biaxial loading

In this paper using finite difference method the lower bound buckling load for simply supported (a) stepped and stiffened rectangular thin plate (b) linear and non-linear variation of thickness (c) uniformly distributed compressive forces in both directions (d) uniformly distributed compressive force in y direction and non-uniform distribution of compressive force in x-direction is discussed. The thin plate is divided into 900 rectangular meshes. The partial derivatives are approximated using central difference formula. Eight hundred and forty one equations are formed and using the program developed and the least eigenvalue is obtained. The buckling coefficients are calculated for different types of stepped and non prismatic plates and the results are presented in tables and graphs for ready use by designers. Buckling factors for some cases are presented in the form of three separate tables and compared with the values obtained by Xiang, Wei and Wang. The results are in close agreement.     

Sharp Reilly-type inequalities for a class of elliptic operators on submanifolds

Let M be an $n(>2)$-dimensional closed orientable submanifold in an $(n+p)$-dimensional space form ${\mathbb{R}}^{n+p}(c)$. We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on M defined by $L_{T}f=-\text{div}(T\mathrm{\nabla }f)$, where T is a general symmetric, positive definite and divergence-free $(1,1)$-tensor on M. The upper bound is given in terms of an integration involving tr T and $|H_T{|}^2$, where tr T is the trace of the tensor T and $H_T={\sum }_{i=1}^{n}A(T{e}_i,e_i)$ is a normal vector field associated with T and the second fundamental form A of M. Furthermore, we give the sufficient and necessary conditions when the upper bound is attained. Our main theorem can be viewed as an extension of the famous “Reilly inequality”. The operator $L_T$ can be regarded as a natural generalization of the well-known operator $L_r$ which is the linearized operator of the first variation of the $(r+1)$-th mean curvature for hypersurfaces in a space form. As applications of our main theorem, we generalize the results of Grosjean [17] and Li–Wang [20] in codimension one to arbitrary codimension.     

On Estimation of Temperature Uncertainty Using the Second Order Adjoint Problem

The uncertainty of temperature prediction from the heat flux error is estimated using first and second order adjoint equations. The adjoint codes developed for the inverse heat transfer problems provide the uncertainty estimation for the corresponding forward problems. Numerical tests corroborate the feasibility of fast uncertainty estimation using Hessian maximum eigenvalue obtained via second order adjoint equations.