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In this paper we study the existence, regularity, radial symmetry and decay property of a mountain pass solution for nonlinear scalar field equations involving the fractional Laplacian under an almost optimal class of continuous nonlinearities.

This paper focuses on the following scalar field equation involving a fractional Laplacian: where N ≥ 2, α ∈ (0, 1), (−Δ)α stands for the fractional Laplacian. Using some minimax arguments, we obtain a positive ground state under the general Berestycki–Lions type assumptions.

The S-wave Klein-Gordon equation with vector and scalar Wood-Saxon potentials is solved, and the energy levels are given through an equation.

We present the exact solutions of the Klein–Gordon equation with equal scalar and vector ring-shaped potentials.

Exact solutions for the scalar field and the potential function in an exponentially inflationary universe, whose source of the gravitational field is a classical scalar field plus an isotropic fluid, are found.

Many of the topological and dynamical properties of a network are related to its Laplacian spectrum; these properties include network diameter, Kirchhoff index, and mean first-passage time. This paper investigates consensus dynamics in a linear dynamical system with additive stochastic disturbances, which is characterized as network coherence by the Laplacian spectrum. We choose a family of uniform recursive trees as our model, and propose a method to calculate the first- and second-order network coherence. Using the tree structures, we identify a relationship between the Laplacian matrix and Laplacian eigenvalues. We then derive the exact solutions for the reciprocals and square reciprocals of all nonzero Laplacian eigenvalues. We also obtain the scalings of network coherence with network size. The scalings of network coherence of the studied trees are smaller than those of Vicsek fractals and are not related to its fractal dimension.

A vector coupling term apart from the scalar and pseudoscalar terms, is introduced into the nonlocal quark-(anti-)quark interaction in an extended Nambu-Jona-Lasinio model to investigate the vector rho meson mode in the relativistic random phase approximation (RPA). It is shown that a favorable choice of the vector coupling to fit the empirical data has to be the same form as that of the original scalar and pseudoscalar ones but with a necessary negative sign.

The properties of the shape of patterns generated by random growth in a Laplacian field are investigated using a simple model, the smoothed Laplacian model. The relation between this model and more complex systems such as Saffman-Taylor fingers or diffusion-limited aggregates, stands in the recently recognized universality of patterns whose evolution is controlled by the Laplace equation, and in an analytical approximation for the Laplacian field itself. The smoothed Laplacian model allows one to treat analytically the equilibrium of the structure in the 'smoothed' Laplacian field in a mean-field approximation. The results on this model are extended to radial geometry thus providing theoretical insight in the dependence of the shape on the geometry of boundary conditions in which the process takes place and on the parameter that tunes the strength of the Laplacian field in the same way as the parameter eta does for the dielectric-breakdown model. This new picture is compared to numerical simulation results of DLA and DBM and proves to be consistent with them.

We consider the relativistic Schr?dinger equation with a time dependent vector and scalar potential on a bounded cylindrical domain. Using a geometric optics ansatz we establish a logarithmic stability estimate for the recovery of vector and scalar potentials.

Submitted by Manuel Brandão (mbrandao@estsp.ipp.pt) on 2015-10-22T10:29:38Z No. of bitstreams: 1 ART_TenreiroMachado10_2014.pdf: 208244 bytes, checksum: 54de6ea2aafb6a7561a2ac980485c40d (MD5) Approved for entry into archive by Ana Rebelo (amsr@isep.ipp.pt) on 2015-11-19T11:47:32Z (GMT) No. of bitstreams: 1 ART_TenreiroMachado10_2014.pdf: 208244 bytes, checksum: 54de6ea2aafb6a7561a2ac980485c40d (MD5) Made available in DSpace on 2015-11-19T11:47:32Z (GMT). No. of bitstreams: 1 ART_TenreiroMachado10_2014.pdf: 208244 bytes, checksum: 54de6ea2aafb6a7561a2ac980485c40d (MD5) Previous issue date: 2014

The paper is devoted to the localization problem of a set of vector field discontinuities, and also discontinuities of its derivatives. We propose new approaches to the reconstruction of vector field singularities by its ray transforms. Alongside modification the Vainberg operator, for solution of this problem we use certain operators of vector analysis in different combinations.

Details of a decomposition of an arbitrary vector field by scalar potentials are given. Known expansions for divergence-free fields, curl-free fields and transverse fields are shown to be special cases.

Exact regular axisymmetric multi-wormhole solutions of the Einstein-Maxwell scalar field equations are generated from corresponding solutions of the Einstein scalar field equations, and their interpretation as multi-particle systems is discussed.

A detailed calculation of the spectrum of the lowest energy, E0, is presented for vector, axial vector, tensor, and scalar fluctuation modes (still incomplete) in eleven-dimensional supergravity compactified on the squashed 7-sphere. The general correspondence is also given between the generalised Laplacian and the quadratic G-Casimir invariant in an arbitrary homogeneous space G/H.

Static solutions of the Einstein-Maxwell scalar equations are obtained from known vacuum solutions, in which the specific electric charges as well as the specific scalar charge may assume arbitrary values. Two types of long-range scalar fields are considered; one is repulsive, like the Coulomb field, the other is attractive, like the gravity field. The effect of the superposition of these two types of fields is also studied. The solutions presented tend straightforwardly to the corresponding vacuum solutions when the constants associated with the electromagnetic and scalar strengths tend to zero.

By means of a resonance theorem we will establish the existence of functions in (where is an -dimensional region) whose expansion in eigenfunctions of the Laplacian is not Riesz-summable of order if .