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    Method for arbitrary phase transformation by a slab based on transformation optics and the principle of equal optical path

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    The optical path lengths travelled by rays across a wavefront essentially determine the resulting phase front irrespective of the shape of a medium according to the principle of equal optical path. Thereupon we propose a method for the transformation between two arbitrary wavefronts by a slab, i.e. the profile of the spatial separation between the two wavefronts is taken to be transformed to a plane surface. Interestingly, for the mutual conversion between planar and curved wavefronts, the method reduce to an inverse transformation method in which it is the reversed shape of the desired wavefront that is converted to a planar one. As an application, three kinds of phase transformation are realized and it is found that the transformation on phase is able to realize some important properties such as phase reversal or compensation, focusing, and expanding or compressing beams, which are further confirmed by numerical simulations. The slab can be applied to realizing compact electromagnetic devices for which the values of the refractive index or the permittivity and permeability can be high or low, positive or negative, or near zero, depending on the choice of coordinate transformations.Comment: 8 pages, 6 figure

    The enumeration of generalized Tamari intervals

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    Let vv be a grid path made of north and east steps. The lattice TAM(v)\rm{T{\scriptsize AM}}(v), based on all grid paths weakly above vv and sharing the same endpoints as vv, was introduced by Pr\'eville-Ratelle and Viennot (2014) and corresponds to the usual Tamari lattice in the case v=(NE)nv=(NE)^n. Our main contribution is that the enumeration of intervals in TAM(v)\rm{T{\scriptsize AM}}(v), over all vv of length nn, is given by 2(3n+3)!(n+2)!(2n+3)!\frac{2 (3n+3)!}{(n+2)! (2n+3)!}. This formula was first obtained by Tutte(1963) for the enumeration of non-separable planar maps. Moreover, we give an explicit bijection from these intervals in TAM(v)\rm{T{\scriptsize AM}}(v) to non-separable planar maps.Comment: 19 pages, 11 figures. Title changed, originally titled "From generalized Tamari intervals to non-separable planar maps (extended abstract)", submitte

    Conformal Anomalies in Noncommutative Gauge Theories

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    We calculate conformal anomalies in noncommutative gauge theories by using the path integral method (Fujikawa's method). Along with the axial anomalies and chiral gauge anomalies, conformal anomalies take the form of the straightforward Moyal deformation in the corresponding conformal anomalies in ordinary gauge theories. However, the Moyal star product leads to the difference in the coefficient of the conformal anomalies between noncommutative gauge theories and ordinary gauge theories. The β\beta (Callan-Symanzik) functions which are evaluated from the coefficient of the conformal anomalies coincide with the result of perturbative analysis.Comment: 17 pages, Latex, no figures, minor corrections and references added; to appear in Phys. Rev.

    Inverse problem in cylindrical electrical networks

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    In this paper we study the inverse Dirichlet-to-Neumann problem for certain cylindrical electrical networks. We define and study a birational transformation acting on cylindrical electrical networks called the electrical RR-matrix. We use this transformation to formulate a general conjectural solution to this inverse problem on the cylinder. This conjecture extends work of Curtis, Ingerman, and Morrow, and of de Verdi\`ere, Gitler, and Vertigan for circular planar electrical networks. We show that our conjectural solution holds for certain "purely cylindrical" networks. Here we apply the grove combinatorics introduced by Kenyon and Wilson.Comment: 22 pages, 15 figure
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