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Apply style for article title, author, affiliation and email as per stylesheet. Several decades ago, antennas had simple shapes that were described in Euclidean geometry. Nowadays, scientists try to make the structure of fractal geometry for applications in the field of electromagnetism, which has led to the development of new innovative antenna devices. Non-integer dimensional space (NDS) is useful to describe the concept of fractional space in fractal structure for real phenomenon of electromagnetic wave propagation. In this work, we investigate effects of NDS and normalized frequency on modulational instability (MI) gain in lossless left-handed metamaterials (LHM). We derive the nonlinear Schrödindiger equation (NLSE) with non-integer transverse laplacian. By means of linear stability analysis method, MI gain expression is also determined. Different forms of figures are obtained due to the signs of group velocity dispersion (GVD) and defocusing/focusing nonlinearity. We show how the increasing value of the normalized frequency enhances the amplitude as well as the bandwidth of MI gain, and waves are more unstable due to non-integer dimension. The obtained results are new and have a relatively newer application in telecommunication by constructing the fractal-shaped antennas operating in multi-frequency bands.

The requirement of reflection positivity (RP) for Euclidean field theories is considered. This is done for the cases of a scalar field, a higher derivative scalar field theory and the scalar field theory defined on a non-integer dimensional space (NIDS). It is shown that regarding RP, the analytical structure of the corresponding Schwinger functions plays an important role. For the higher derivative scalar field theory, RP does not hold. However for the scalar field theory on a NIDS, RP holds in a certain range of dimensions where the corresponding Minkowskian field is defined on a Hilbert space with a positive definite scalar product that provides a unitary representation of the Poincaré group. In addition, and motivated by the last example, it is shown that, under certain conditions, one can construct non-local reflection positive Euclidean field theories starting from the corrected two point functions of interacting local field theories.