Congruence Lattices of Certain Finite Algebras with Three Commutative Binary Operations

A partial algebra construction of Gr\"atzer and Schmidt from "Characterizations of congruence lattices of abstract algebras" (Acta Sci. Math. (Szeged) 24 (1963), 34-59) is adapted to provide an alternative proof to a well-known fact that every finite distributive lattice is representable, seen as a special case of the Finite Lattice Representation Problem. The construction of this proof brings together Birkhoff's representation theorem for finite distributive lattices, an emphasis on boolean lattices when representing finite lattices, and a perspective based on inequalities of partially ordered sets. It may be possible to generalize the techniques used in this approach. Other than the aforementioned representation theorem only elementary tools are used for the two theorems of this note. In particular there is no reliance on group theoretical concepts or techniques (see P\'eter P\'al P\'alfy and Pavel Pud\'lak), or on well-known methods, used to show certain finite lattice to be representable (see William J. DeMeo), such as the closure method

New Angle on the Strong CP and Chiral Symmetry Problems from a Rotating Mass Matrix

It is shown that when the mass matrix changes in orientation (rotates) in generation space for changing energy scale, then the masses of the lower generations are not given just by its eigenvalues. In particular, these masses need not be zero even when the eigenvalues are zero. In that case, the strong CP problem can be avoided by removing the unwanted $\theta$ term by a chiral transformation in no contradiction with the nonvanishing quark masses experimentally observed. Similarly, a rotating mass matrix may shed new light on the problem of chiral symmetry breaking. That the fermion mass matrix may so rotate with scale has been suggested before as a possible explanation for up-down fermion mixing and fermion mass hierarchy, giving results in good agreement with experiment.Comment: 14 page

Plasmonic modes of polygonal particles calculated using a quantum hydrodynamics method

Plasmonic resonances of nanoparticles have drawn lots of attentions due to their interesting and useful properties such as strong field enhancements. These systems are typically studied using either classical electrodynamics or fully quantum theory. Each approach can handle some aspects of plasmonic systems accurately and efficiently, while having its own limitation. The self-consistent hydrodynamics model has the advantage that it can incorporate the quantum effect of the electron gas into classical electrodynamics in a consistent way. We use the method to study the plasmonic response of polygonal particles under the influence of an external electromagnetic wave, and we pay particular attention to the size and shape of the particle and the effect of charging. We find that the particles support edge modes, face modes and hybrid modes. The charges induced by the external field in the edge (face) modes mainly localize at the edges (faces), while the induced charges in the hybrid modes are distributed nearly evenly in both the edges and faces. The edge modes are less sensitive to particle size than the face modes, but are sensitive to the corner angles of the edges. When the number of sides of regular polygons increases, the edge and face modes gradually change into the classical dipole plasmonic mode of a cylinder. The hybrid modes are found to be the precursor of the Bennett mode, which cannot be found in classical electrodynamics.Comment: 33 pages, 9 figure

Vortex-like surface wave and its role in the transient phenomena of meta-material focusing

We show that a slab of meta-material (with $\epsilon=\mu=-1+i\Delta$) possesses a vortex-like surface wave with no ability to transport energy, whose nature is completely different from a localized mode or a standing wave. Through computations based on a rigorous time-dependent Green's function approach, we demonstrate that such a mode inevitably generates characteristic image oscillations in two dimensional focusing with even a monochromatic source, which were observed in many numerical simulations, but such oscillations are weak in three dimensional focusing.Comment: To appear in the March 7th issue of Appl. Phys. Let