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Register a new userClassification over a predicate -- the general case. Part I -- structure theory
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We begin the development of structure theory for a first order theory stable over a monadic predicate.
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The theoretical interpretation of measurements of wavefunctions and spectra in electromagnetic cavities excited by antennas is considered. Assuming that the characteristic wavelength of the field inside the cavity is much larger than the radius of the antenna, we describe antennas as point-like perturbations. This approach strongly simplifies the problem reducing the whole information on the antenna to four effective constants. In the framework of this approach we overcame the divergency of series of the phenomenological scattering theory and justify assumptions lying at the heart of wavefunction measurements. This selfconsistent approach allowed us to go beyond the one-pole approximation, in particular, to treat the experiments with degenerated states. The central idea of the approach is to introduce ``renormalized Green function, which contains the information on boundary reflections and has no singularity inside the cavity.
A logic satisfies the interpolation property provided that whenever a formula {Delta} is a consequence of another formula {Gamma}, then this is witnessed by a formula {Theta} which only refers to the language common to {Gamma} and {Delta}. That is, the relational (and functional) symbols occurring in {Theta} occur in both {Gamma} and {Delta}, {Gamma} has {Theta} as a consequence, and {Theta} has {Delta} as a consequence. Both classical and intuitionistic predicate logic have the interpolation property, but it is a long open problem which intermediate predicate logics enjoy it. In 2013 Mints, Olkhovikov, and Urquhart showed that constant domain intuitionistic logic does not have the interpolation property, while leaving open whether predicate Godel logic does. In this short note, we show that their counterexample for constant domain intuitionistic logic does admit an interpolant in predicate Godel logic. While this has no impact on settling the question for predicate Godel logic, it lends some credence to a common belief that it does satisfy interpolation. Also, our method is based on an analysis of the semantic tools of Olkhovikov and it is our hope that this might eventually be useful in settling this question.
We study the R-torsionfree part of the Ziegler spectrum of an order Lambda over a Dedekind domain R. We underline and comment on the role of lattices over Lambda. We describe the torsionfree part of the spectrum when Lambda is of finite lattice representation type.
In this paper, we use some basic quasi-topos theory to study two functors: one adding infinitesimals of Fermat reals to diffeological spaces (which generalize smooth manifolds including singular spaces and infinite dimensional spaces), and the other deleting infinitesimals on Fermat spaces. We study the properties of these functors, and calculate some examples. These serve as fundamentals for developing differential geometry on diffeological spaces using infinitesimals in a future paper.
We prove that the theory of all modules over the ring of algebraic integers is decidable.
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