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In their simplest formulations, classical dynamics is the study of Hamiltonian flows and quantum mechanics of propagators. Both are linked, and emerge from the datum of a single classical concept, the Hamiltonian function. We study and emphasize the analogies between Hamiltonian flows and quantum propagators; this allows us to verify G. Mackey's observation that quantum mechanics (in its Weyl formulation) is a refinement of Hamiltonian mechanics. We discuss in detail the metaplectic representation, which very explicitly shows the close relationship between classical mechanics and quantum mechanics; the latter emerging from the first by lifting Hamiltonian flows to the double covering of the symplectic group. We also give explicit formulas for the factorization of Hamiltonian flows into simpler flows, and prove a quantum counterpart of these results.

We give a list of possibilities for surfaces of general type with p_g = 0 having an involution i such that the bicanonical map of S is not composed with i and S/i is not rational. Some examples with K² = 4, …, 7 are constructed as double coverings of an Enriques surface. These surfaces have a description as bidouble coverings of the plane.

Twisted links are a generalization of classical links and correspond to stably equivalence classes of links in thickened surfaces. In this paper, we introduce twisted intersection colorings of a diagram and construct two invariants of a twisted link using such colorings. As an application, we show that there exist infinitely many pairs of twisted links such that for each pair the two twisted links are not equivalent but their double coverings are equivalent.

We also introduce a method of constructing a pair of twisted links whose double coverings are equivalent.

Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the notion of the double covering of a twisted link. It is defined by considering the orientation double covering of an abstract link or alternatively by constructing a diagram called a double covering diagram. We also discuss links in thickened surfaces, their diagrams and their stable equivalence classes. Bourgoin’s twisted knot group is understood as the virtual knot group of the double covering.

The spectra of generalized Cayley graphs of finite abelian groups are investigated in this paper. For a generalized Cayley graph $X$ of a finite group $G$, the canonical double covering of $X$ is the direct product $X\times K_2$. In this paper, integral generalized Cayley graphs on finite abelian groups are characterized, using the characterization of the spectra of integral Cayley graphs. As an application, the integral generalized Cayley graphs on ${\mathbb{Z}}_p\times {\mathbb{Z}}_q$ and ${\mathbb{Z}}_{2^n}$ are investigated, where $p$ and $q$ are odd prime numbers.

In this paper, using the theory of double coverings of cyclotomic fields, we give a formula for , where K = ℚ(ζ_n), G = Gal(K/ℚ), 𝔽₂ = ℤ/2ℤ and U_K is the unit group of K. We explicitly determine all the cyclotomic fields K = ℚ(ζ_n) such that . Then we apply it to the unit square problem raised in [Y. Li and X. Zhang, Global unit squares and local unit squares, J. Number Theory128 (2008) 2687–2694]. In particular, we prove that the unit square problem does not hold for ℚ(ζ_n) if n has more than three distinct prime factors, i.e. for each odd prime p, there exists a unit, which is a square in all local fields ℚ(ζ_n)_v with v | p but not a square in ℚ(ζ_n), if n has more than three distinct prime factors.