Examples and Counterexamples最新文献

This work studies mathematical foundations for optimal boundary control of mixers which mix two fluid phases. Existence of optima is proved and the influence of common objective functionals is discussed.

Amorphous silicon is a highly promising anode material for next-generation lithium-ion batteries. Large volume changes of the silicon particle have a critical effect on the surrounding solid-electrolyte interphase (SEI) due to repeated fracture and healing during cycling. Based on a thermodynamically consistent chemo-elasto-plastic continuum model we investigate the stress development inside the particle and the SEI. Using the example of a particle with SEI, we apply a higher order finite element method together with a variable-step, variable-order time integration scheme on a nonlinear system of partial differential equations. Starting from a single silicon particle setting, the surrounding SEI is added in a first step with the typically used elastic Green–St-Venant (GSV) strain definition for a purely elastic deformation. For this type of deformation, the definition of the elastic strain is crucial to get reasonable simulation results. In case of the elastic GSV strain, the simulation aborts. We overcome the simulation failure by using the definition of the logarithmic Hencky strain. However, the particle remains unaffected by the elastic strain definitions in the particle domain. Compared to GSV, plastic deformation with the Hencky strain is straightforward to take into account. For the plastic SEI deformation, a rate-independent and a rate-dependent plastic deformation are newly introduced and numerically compared for three half cycles for the example of a radial symmetric particle.

The problem of establishing an upper bound for the volume of a parallelepiped is considered by utilizing an original approach involving a skew-symmetric matrix of order four (along with its Moore–Penrose inverse). It is shown that the commonly known inequality characterizing the bound can be virtually sharpened. Similarly, a sharpening is established with respect to the Cauchy–Schwarz inequality. General properties of the Moore–Penrose inverse of a skew-symmetric matrix are discussed as well.

In this work, we first introduce the two-sided quaternion Fourier transform and demonstrate its essential properties. We generalize Titchmarsh’s-type theorem in the framework of the two-sided quaternion Fourier transform. Based on the interaction between the quaternion Fourier transform and quaternion linear canonical transform we explore sharp Hausdorff–Young inequality for the quaternion linear canonical transform. The obtained result can be considered as a generalized version of sharp Hausdorff–Young inequality for the two-dimensional quaternion Fourier transformation in the literature.

This paper compares the notions of Auslander generalized recurrence and chain recurrence of dynamical systems. Generalized recurrent points are chain recurrent, however, chain recurrent points may not be generalized recurrent of any order. Because of the absence of recursiveness, the dispersive systems have no generalized recurrent point. Examples of systems with generalized recurrent points and systems with no generalized recurrent points are presented. The main example shows a dispersive system that is chain transitive.

Locating the origin of diffusion in complex networks is an interesting but challenging task. It is crucial for anticipating and constraining the epidemic risks. Source localization has been considered under many feasible models. In this paper, we study the localization problem in some path-related graphs and study the edge metric dimension. A subset $L_E\subseteq V_G$ is known as an edge metric generator for $G$ if, for any two distinct edges $e_1,e_2\in E$, there exists a vertex $a\subseteq L_E$ such that $d(e_1,a)\ne d(e_2,a)$. An edge metric generator that contains the minimum number of vertices is termed an edge metric basis for $G$, and the number of vertices in such a basis is called the edge metric dimension, denoted by $di{m}_e\left(G\right)$. An edge metric generator with the fewest vertices is called an edge metric basis for $G$. The number of vertices in such a basis is the edge metric dimension, represented as $di{m}_e\left(G\right)$. In this paper, the edge metric dimension of some path-related graphs is computed, namely, the middle graph of path $M\left(P_n\right)$ and the splitting graph of path $S\left(P_n\right)$.