We consider a two-dimensional atomic mass spring system and show that in the small displacement regime the corresponding discrete energies can be related to a continuum Griffith energy functional in the sense of Γ-convergence. We also analyze the continuum problem for a rectangular bar under tensile boundary conditions and find that depending on the boundary loading the minimizers are either ho...

We prove the stability of a large class of unilateral minimality properties which arise naturally in the theory of crack propagation proposed by Francfort and Marigo in [22]. Then we give an application to the quasistatic evolution of cracks in composite materials.

We give results for the Γ-limit of a scaled elastic energy of a film as the thickness h > 0 converges to zero. The elastic energy density models materials with multiple phases or variants and is thus non-convex. The model includes an interfacial energy that allows sharp interfaces between the phases and variants and is proportional to the total variation of the deformation gradient.

Theorem 1.1 (Popa’s Cocycle Superrigity, Special Case). Let Γ be a discrete group with Kazhdan’s property (T), let Γ0 < Γ be an infinite index subgroup, (X0,μ0) an arbitrary probability space, and let Γ (X,μ) = (X0,μ0)0 be the corresponding generalized Bernoulli action. Then for any discrete countable group Λ and any measurable cocycle α : Γ × X → Λ there exist a homomorphism : Γ → Λ and a meas...

background: periodontitis is an inflammatory disease of tooth-supporting tissues. the present study was conducted to evaluate interleukin-10 (il-10) and interferon gamma (ifn-γ) levels in gingival crevicular fluid (gcf) of patients with chronic periodontitis. materials and methods: this case-control study was carried out on 60 men including 30 smokers and 30 age-matched non-smokers. the assessm...

Let Σ g,b be an oriented genus g surface with b boundary components and p punctures and let Mod(Σ g,b) be its mapping class group, that is, the group of isotopy classes of orientation–preserving diffeomorphisms of Σ g,b that fix the boundary components and punctures pointwise (we will omit b or p if they vanish). A long–standing conjecture of Ivanov (see [7] for a recent discussion) says that f...

the performance of a γ-al 2 o 3 /n-decane nanofluid shell-and-tube heat exchanger in a biomass heating plant is analyzed to specify the optimum condition based on the maximum heat transfer rate and performance index for wide range of nanoparticle volume fraction (0–7%). compared with pure          n-decane, the obtained results in this research show that by using γ-al 2 o 3 /n-decane nanofluid ...

The aim of this article to follow the properties zero-divisor graph special idealization ring. We study wiener index zero-divisors some ring R(+)M and find clique number Γ(R(+)M) is ω (Γ(R(+)M)) = |M| − 1, where R an integral domain. also discuss when are Hamiltonian graph.

γ uLγ u where Lγ u = u ′′ +K u, the Morse index is the number of negative eigenvalues of Lγ. (By convention, an eigenfunction φ with eigenvalue λ of Lγ is a solution of Lγ φ+ λφ = 0.) Note that if λ = 0, then φ (or φn) is a (normal) Jacobi field. γ is stable if the index is zero. The index of a noncompact geodesic is the dimension of a maximal vector space of compactly supported variations for ...