We study the asymptotic behavior, when $$\varepsilon \rightarrow 0$$ , of minimizers $$\{u_\varepsilon \}_{\varepsilon >0}$$ for energy $$\begin{aligned} E_\varepsilon (u)=\int _{\Omega }\Big (|\nabla u|^2+\big (\frac{1}{\varepsilon ^2}-1\big )|\nabla |u||^2\Big ), \end{aligned}$$ over class maps $$u\in H^1(\Omega ,{{\mathbb {R}}}^2)$$ satisfying boundary condition $$u=g$$ on $$\partial \Omega ...

We prove a theorem that can be thought of as common generalization the Discrete Nodal Theorem and (one direction of) Cheeger's Inequality for graphs. A special case this result will assert if second third eigenvalues Laplacian are at least $\varepsilon$ apart, then subgraphs induced by positive negative supports eigenvector belonging to $\lambda_2$ not only connected, but edge-expanders (in wei...

We present a systemic theoretical study of the electronic properties of the quantum dots inserted in quantum dot infrared photodetectors (QDIPs). The strain distribution of three different shaped quantum dots (QDs) with a same ratio of the base to the vertical aspect is calculated by using the short-range valence-force-field (VFF) approach. The calculated results show that the hydrostatic strai...

The dynamics of a globally coupled, logistic map lattice is explored over a parameter plane consisting of the coupling strength, varepsilon, and the map parameter, a. By considering simple periodic orbits of relatively small lattices, and then an extensive set of initial-value calculations, the phenomenology of solutions over the parameter plane is broadly classified. The lattice possesses many...

This paper proves a tradeoff between the time it takes to search for elements in an implicit dictionary and the time it takes to update the value of elements in specified locations of the dictionary. It essentially shows that if the update time is constant, then the search time is $\Omega(n’)$ for some constant $\varepsilon >0$.

Given a closed manifold M and connected submanifold \(N\subset M\) of positive codimension, we study the Steklov spectrum domain \(\varOmega _\varepsilon \subset obtained by removing tubular neighborhood size \(\varepsilon \) around N. All nonzero eigenvalues in mid-frequency range tend to infinity at rate which depends only on codimension N M. Eigenvalues above are also described: they followi...

We offer a new definition of $varepsilon$-orthogonality in normed spaces, and we try to explain some properties of which. Also we introduce some types of $varepsilon$-orthogonality in an arbitrary  $C^ast$-algebra $mathcal{A}$, as a Hilbert $C^ast$-module over itself, and investigate some of its properties in such spaces. We state some results relating range-kernel orthogonality in $C^*$-algebras.