Crystalline topological phases have recently attracted a lot of experimental and theoretical attention. Key advances include the complete elementary band representation analyses of crystalline matter by symmetry indicators and the discovery of higher-order hinge and corner states. However, current classification schemes of such phases are either implicit or limited in scope. We present a new scheme for the explicit classification of crystalline topological insulators and superconductors. These phases are protected by crystallographic point group symmetries and are characterized by bulk topological invariants. The classification paradigm generalizes the Clifford algebra extension process of each Altland-Zirnbauer symmetry class and utilizes algebras which incorporate the point group symmetry. Explicit results for all point group symmetries of three-dimensional crystals are presented as well as for all symmorphic layer groups of two-dimensional crystals. We discuss future extensions for treatment of magnetic crystals and defected or higher-dimensional systems as well as weak and fragile invariants.

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Arrays of dielectric resonators-illuminated by an antenna-are used to ignite and sustain multiple microwave plasmas in parallel. Calcium titanate cylindrical resonators were arranged in a linear array with separation distances between 0.5 and 5 mm. The operating frequency was near the HEM 111 resonance of 1.1 GHz. Paschen curves of the breakdown field and voltage in argon atmosphere are consistent with parallel plate microwave breakdown except within discharge gaps of 1 mm or less. Sustaining of argon plasma between 0.5 Torr and 1 atm within the array is found to alter the electromagnetic scattering from the dielectric resonators, suggesting applications in plasma-reconfigurable metamaterials and photonic crystals.

We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings D. We define their fixed points to be the points $\lambda \in D$ for which $f^{\circ n}(\lambda )=\lambda $ for any $n \in \mathbb {N}$ , where $f^{\circ n}(x)$ is defined recursively by $f^{\circ n}(x)=f(f^{\circ (n-1)}(x))$ and $f^{\circ 1}(x)=f(x)$ . Periodic points are similarly defined. We prove that $\lambda $ is a fixed point of $f(x)$ if and only if $f(\lambda )=\lambda $ , which enables the use of known results from the theory of polynomial equations, to conclude that any polynomial of degree $m \geq 2$ has at most m conjugacy classes of fixed points. We also show that in general, periodic points do not behave as in the commutative case. We provide a sufficient condition for periodic points to behave as expected.

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In this paper, we present a complete method for finding the roots of all polynomials of the form [Formula: see text] over a given octonion division algebra. When [Formula: see text] is monic, we also consider the companion matrix and its left and right eigenvalues and study their relations to the roots of [Formula: see text], showing that the right eigenvalues form the conjugacy classes of the roots of [Formula: see text] and the left eigenvalues form a larger set than the roots of [Formula: see text].

We show that if two division p-algebras of prime degree share an inseparable field extension of the center then they also share a cyclic separable one. We show that the converse is in general not true. We also point out that sharing all the inseparable field extensions of the center does not imply sharing all the cyclic separable ones.