The field equations of Brans-Dicke gravity coupled to a mass-varying vector field are derived. Anisotropic cosmological solutions with a locally rotationally symmetric Bianchi type I metric and time-dependent scalar and electric vector fields are studied. A particular class of exact solutions for which all the variable parameters have a power-law time dependence is given. The universe expands with a constant expansion anisotropy within this class of solutions. We show that the accelerating expansion is driven by the scalar field and the electric vector field can be interpreted as an anisotropic dark matter source.

A nonrelativistic quantum mechanical particle moving freely on a curved surface feels the effect of the nontrivial geometry of the surface through the kinetic part of the Hamiltonian, which is proportional to the Laplace-Beltrami operator, and a geometric potential, which is a linear combination of the mean and Gaussian curvatures of the surface. The coefficients of these terms cannot be uniquely determined by general principles of quantum mechanics but enter the calculation of various physical quantities. We examine their contribution to the geometric scattering of a scalar particle moving on an asymptotically flat embedded surface. In particular, having in mind the possibility of an experimental realization of the geometric scattering in a low density electron gas formed on a bumped surface, we determine the scattering amplitude for arbitrary choices of the curvature coefficients for a surface with global or local cylindrical symmetry. We also examine the effect of perturbations that violate this symmetry and consider surfaces involving bumps that form a lattice.

A higher dimensional modified gravity theory with an action that includes dimensionally continued Euler-Poincaré forms up to second order in curvatures is considered. The variational field equations are derived. Matter in the universe at large scales is modeled by a fluid satisfying an equation of state with dimensional dichotomy. We study solutions that describe higher dimensional steady state cosmologies with constant volume for which the three dimensional external space is expanding at an accelerated rate while the (compact) internal space is contracting. We showed that the second order Euler-Poincaré term in the constructions of higher dimensional steady state cosmologies could be crucial.

We give explicit criteria for the reflectionlessness, transparency, and invisibility of a finite-range potential in the presence of an incoherent (intensity-dependent) nonlinearity that is confined to the range of the potential. This allows us to conduct a systematic study of the effects of such a nonlinearity on a locally periodic class of finite-range potentials that display perturbative unidirectional invisibility. We use our general results to examine the effects of a weak Kerr nonlinearity on the behavior of these potentials and show that the presence of nonlinearity destroys the unidirectional invisibility of these potentials. If the strength of the Kerr nonlinearity is so weak that the first-order perturbation theory is reliable, the presence of nonlinearity does not affect the unidirectional reflectionlessness and transmission reciprocity of the potential. We show that the expected violation of the latter is a second order perturbative effect.Keywords: Unidirectional invisibility, complex potential, reflectionless potential, nonlinear scattering, incoherent nonlinearity, Kerr nonlinearity, reciprocity principle An important difference between real and complex scattering potentials is that the reciprocity in reflection is generally broken for a complex potential. An extreme example is a potential v(x) that is reflectionless only from the left or right. If, in addition, v(x) has perfect transmission property, i.e., the transmitted (left-or right-going) waves are not affected by the presence of v(x), it is said to be unidirectionally invisible [1,2,3]. The aim of the present letter is to explore the consequences of introducing a weak incoherent nonlinearity on the behavior of a unidirectionally invisible potential. 1Consider an isotropic nonmagnetic medium with translational symmetry along the y-and z-axes. The interaction of this medium with electromagnetic waves is described by its relative permittivityε(x). A normally incident z-polarized TE wave that propagates in such a medium has an electric field of the form E( r, t) = E 0 e −ikct ψ(x)ê z , where r is the position vector, E 0 * Corresponding author, Email Address: amostafazadeh@ku.edu.tr 1 Here by "incoherent" we mean that the nonlinear term in the wave equation does not depend on the phase of its solutions ψ, i.e., it is a function of |ψ|.

We show that the Helmholtz equation describing the propagation of transverse electric waves in a Kerr slab with a complex linear permittivity and a complex Kerr coefficient admits blow-up solutions provided that the real part of the Kerr coefficient is negative, i.e., the slab is defocusing.This result applies to homogeneous as well as inhomogeneous Kerr slabs whose linear permittivity and Kerr coefficient are continuous functions of the transverse coordinate. For an inhomogeneous Kerr slab, blow-up solutions exist if the real part of Kerr coefficient is bounded above by a negative number. *