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All About Wave Equations
Published in Bahman Zohuri, Patrick J. McDaniel, Electrical Brain Stimulation for the Treatment of Neurological Disorders, 2019
Bahman Zohuri, Patrick J. McDaniel
Note: There are couple experiments proposed by Raymond C. Gelinas,34 in respect to a demonstration of CFVP in a simply connected space, which involves a toroidal coil carrying current the provides a source of CFVP which extends over a large region of space. Thus, the reader who are interested to further investigate this subject should follow his approach and suggestion and it is beyond the scope of this book.
Eigenvalue inequalities for the p-Laplacian operator on C-totally real submanifolds in Sasakian space forms
Published in Applicable Analysis, 2022
Akram Ali, Ali H. Alkhaldi, Pişcoran Laurian-Ioan, Rifaqat Ali
Using the Wentzel–Laplacian operator on a compact submanifold with boundary in the Euclidean space, Du et al. [9] provided an estimation of the first nonzero eigenvalue. After that, using a decomposition of the Hessian on Kaehler manifolds with positive Ricci curvature, Blacker and Seto [10] proved a Lichnerowicz lower bound theorem for the first nonzero eigenvalue of the p-Laplacian on Kaehler manifold, for Dirichlet and Neumann boundary conditions. Furthermore, a submanifold can be immersed in an m-dimensional simply connected space form of constant sectional curvature c which included the Euclidean space , the unit sphere and the hyperbolic space with c = 0, 1 and respectively. Such estimation for the first nonzero eigenvalue of the Laplacian, has been proved in [1,11]. It well known, all the results obtained until now for several classes of Riemannian submanifolds in some different ambient spaces, show that, both the first nonzero eigenvalues (Dirichlet or Neumann) satisfies similar inequalities and consequently they have identical upper bounds [11,12]. Several great successes in Riemannian geometry were obtained for the theory of p-Laplacians on Riemannian submanifolds in different ambient manifolds (see [4,5,12–20]) and also through the work [3]. Inspired by this notion, our method is based on the derivation of the first eigenvalue for the p-Laplacian on C-totally real submanifold of Sasakian space form. From this point of view, using the Gauss equation and [1,14], we have been motivated the study of the first nonzero eigenvalue of the p-Laplacian on submanifold in different space forms. We proved a sharp upper bound theorem for the first eigenvalue for the p-Laplacian operator on C-totally real submanifold of Sasakian space form (cf. Theorem 3.2).
Micropolar continua as projective space of Skyrmions
Published in Philosophical Magazine, 2022
In a sequential representation of a fibre F, a total space E and its projected base space M of , we can express real and complex projective spaces using the Hopf fibrations. We write some of the important fibrations for n-dimensional spheres as follows Particular interests arise when we consider the homotopy group relation on these fibrations. For example, suppose that the given manifold M is simply connected. Then any simple closed-loop contained in the given manifold can be continuously deformed into another loop and eventually can be deformed to a point. Then, by definition of the fundamental group, we will have a trivial homotopy . Since all , are simply connected, while for are not, we have for . Moreover, by the Lifting Properties of the fundamental group [36] between the non-simply connected space and its universal covering space, there is an isomorphism This will be useful when one considers the homotopy of an order parameter space . Specifically, an order parameter spaceM can be regarded as an image of a function for , As the system undergoes some phase transitions, either by an external factor or spontaneously, the symmetry G in M may be altered to be its subgroup H. Consequently, there may be regions where the degrees of the order are not uniquely defined. These regions are characterised by a modified quotient group G/H. These regions are called the defects and the names of defect with respective dimension d are (i) monopole: a point-like defect in d = 0, (ii) vortex: a string-like defect in d = 1, (iii) domain wall: a sheet-like defect in d = 2. These defects can be understood in connection with topological invariant quantities and can be found in diverse physical systems with order parameters describing the defects of distinct nature [10, 37–42]. In [43] the connection between the phase transitions that originated from the spontaneous symmetry breaking and those based on the topological nature is studied. These topological invariants are the classification of the defects for a given dimension belonging to one of the equivalence classes given by the homotopy group of the order parameter space M. This means that the homotopy classification determines the allowed range of configurations to be deformed continuously within the given equivalence class.