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The goal of this paper is to report on a formalization of the p-adic numbers in the setting of the second author's univalent foundations program. This formalization, which has been verified in the Coq proof assistant, provides an approach to the p-adic numbers in constructive algebra and analysis.
Mechanical properties of parts constructed with additive manufacturing (AM) technologies are highly influenced by raw material and process characteristics. It is widely assumed that a certain degree of anisotropy should be expected in AM parts due to their layer-upon-layer nature. Present work focuses on the PolyJet process, where each layer is built by selective jetting of photopolymers upon flat surfaces and subsequent UV radiation curing. An extensive experimental program was carried out to find out if the so-constructed parts present viscoelastic behavior and if their mechanical characteristics also depend on part orientation. Both hypotheses have been proven true, so a viscoelastic orthotropic-like behavior shall be expected in PolyJet manufactured part. Nevertheless, a significant improvement on material properties has been found for nearly vertical building orientations. This unexpected behavior is related to a shielding effect upon UV curing caused by support material.
The study of fixed points is a classical subject in geometry and dynamics. If the circle acts in a Hamiltonian fashion on a compact symplectic manifold M, then it is classically known that there are at least fixed points; this follows from Morse theory for the momentum map of the action. In this paper we use Atiyah–Bott–Berline–Vergne (ABBV) localization in equivariant cohomology to prove that this conclusion also holds for symplectic circle actions with non-empty fixed sets, as long as the Chern class map is somewhere injective—the Chern class map assigns to a fixed point the sum of the action weights at the point. We complement this result with less sharp lower bounds on the number of fixed points, under no assumptions; from a dynamical systems viewpoint, our results imply that there is no symplectic periodic flow with exactly one or two equilibrium points on a compact manifold of dimension at least eight.
In this note we describe the natural coordinatizations of a Delzant space defined as a reduced phase space (symplectic geometry view-point) and give explicit formulas for the coordinate transformations. For each fixed point of the torus action on the Delzant polytope, we have a maximal coordinatization of an open cell in the Delzant space which contains the fixed point. This cell is equal to the domain of definition of one of the natural coordinatizations of the Delzant space as a toric variety (complex algebraic geometry view-point), and we give an explicit formula for the toric variety coordinates in terms of the reduced phase space coordinates. We use considerations in the maximal coordinate neighborhoods to give simple proofs of some of the basic facts about the Delzant space, as a reduced phase space, and as a toric variety. These can be viewed as a first application of the coordinatizations, and serve to make the presentation more self-contained.
We present the main results of a study of the observed internal structure constants, k2, for a wide set of eclipsing binaries. From the analysis of the variations in relative positions of the eclipses and the comparison with different theoretical models, we could deduce that the discrepancy, previously reported by several authors between theory and observations, is no longer supported. Moreover, a strong correlation has been found between the evolution of the parameter k2 and the gravity at the surface of the star, g.
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