624,102 research outputs found
The Structure of Models of Second-order Set Theories
This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of T-realizations of a fixed countable model of ZFC, where T is a reasonable second-order set theory such as GBC or KM, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve the existence/nonexistence of upper bounds, at least for finite partial orders. Second I generalize some constructions of Marek and Mostowski from KM to weaker theories. They showed that every model of KM plus the Class Collection schema âunrollsâ to a model of ZFCâ with a largest cardinal. I calculate the theories of the unrolling for a variety of second-order set theories, going as weak as GBC + ETR. I also show that being T-realizable goes down to submodels for a broad selection of second-order set theories T. Third, I show that there is a hierarchy of transfinite recursion principles ranging in strength from GBC to KM. This hierarchy is ordered first by the complexity of the properties allowed in the recursions and second by the allowed heights of the recursions. Fourth, I investigate the question of which second-order set theories have least models. I show that strong theoriesâsuch as KM or Î 11-CAâdo not have least transitive models while weaker theoriesâfrom GBC to GBC + ETROrd âdo have least transitive models
The Structure of Models of Second-order Set Theories
This dissertation is a contribution to the project of second-order set
theory, which has seen a revival in recent years. The approach is to understand
second-order set theory by studying the structure of models of second-order set
theories. The main results are the following, organized by chapter. First, I
investigate the poset of -realizations of a fixed countable model of
, where is a reasonable second-order set theory such as
or , showing that it has a rich structure. In
particular, every countable partial order embeds into this structure. Moreover,
we can arrange so that these embedding preserve the existence/nonexistence of
upper bounds, at least for finite partial orders. Second I generalize some
constructions of Marek and Mostowski from to weaker theories.
They showed that every model of plus the Class Collection schema
"unrolls" to a model of with a largest cardinal. I calculate
the theories of the unrolling for a variety of second-order set theories, going
as weak as . I also show that being -realizable
goes down to submodels for a broad selection of second-order set theories .
Third, I show that there is a hierarchy of transfinite recursion principles
ranging in strength from to . This hierarchy is
ordered first by the complexity of the properties allowed in the recursions and
second by the allowed heights of the recursions. Fourth, I investigate the
question of which second-order set theories have least models. I show that
strong theories---such as or ---do
not have least transitive models while weaker theories---from to
---do have least transitive models.Comment: This is my PhD dissertatio
Anomalous dimensions of potential top-partners
We discuss anomalous dimensions of top-partner candidates in theories of
Partial Compositeness. First, we revisit, confirm and extend the computation by
DeGrand and Shamir of anomalous dimensions of fermionic trilinears. We present
general results applicable to all matter representations and to composite
operators of any allowed spin. We then ask the question of whether it is
reasonable to expect some models to have composite operators of sufficiently
large anomalous dimension to serve as top-partners. While this question can be
answered conclusively only by lattice gauge theory, within perturbation theory
we find that such values could well occur for some specific models. In the
Appendix we collect a number of practical group theory results for fourth-order
invariants of general interest in gauge theories with many irreducible
representations of fermions.Comment: 21 pages, 4 figures, 6 tables V2: Added Table 3,4,5, equation (9) and
various comments in reply to questions and suggestions raised by the two
Referees of SciPost. Two references also added. V3: Typo in footnote 6
corrected. Final version in SciPos
Strain gradient continuum mechanics: simplified models, variational formulations and isogeometric analysis with applications
This dissertation is devoted to two families of generalized continuum theories: the first and second strain gradient elasticity theories including the first and second velocity gradient inertia, respectively. First of all, a number of model problems is studied by analytical means revealing the key characters and potential of generalized continuum theories. In particular, the classical Kirsch problem is extended to the case of a simplified first strain gradient elasticity model demonstrating the size dependency of stresses and strains in the vicinity of a round hole in a plate in tension. Within linearly isotropic second strain gradient elasticity theory, instead, a simplified model is proposed, still capable of capturing free surface effects and surface tension, in particular, arising in solids of both nano- and macro-scales. With a series of benchmark problems, including a comprehensive set of stability analyses, the role of higher-order material parameters is revealed. On the way towards computational analysis, the boundary value problems of the fourth- and sixth-order partial differential equations arising in the first and second strain gradient models, respectively, are formulated and analysed in a mathematical variational form within appropriate Sobolev space settings. For numerical simulations, isogeometric Galerkin methods meeting higher-order continuity requirements are implemented in a user element framework of a commercial finite element software. Various benchmarks for statics and free vibrations confirm the optimal convergence properties of the numerical methods, verify the implementation and demonstrate the key properties of the underlying higher-order continuum models. Regarding model validation and applications, thorough analyses of stretching, shearing and vibration phenomena of complex triangular lattices homogenized by the simplified second strain gradient elasticity model reveal the strong size dependency of lattice structures and hence provide pivotal information for practical applications of materials and structures with a microstructure or microarchitecture
The algebra of Wick polynomials of a scalar field on a Riemannian manifold
On a connected, oriented, smooth Riemannian manifold without boundary we
consider a real scalar field whose dynamics is ruled by , a second order
elliptic partial differential operator of metric type. Using the functional
formalism and working within the framework of algebraic quantum field theory
and of the principle of general local covariance, first we construct the
algebra of locally covariant observables in terms of equivariant sections of a
bundle of smooth, regular polynomial functionals over the affine space of the
parametrices associated to . Subsequently, adapting to the case in hand a
strategy first introduced by Hollands and Wald in a Lorentzian setting, we
prove the existence of Wick powers of the underlying field, extending the
procedure to smooth, local and polynomial functionals and discussing in the
process the regularization ambiguities of such procedure. Subsequently we endow
the space of Wick powers with an algebra structure, dubbed E-product, which
plays in a Riemannian setting the same role of the time ordered product for
field theories on globally hyperbolic spacetimes. In particular we prove the
existence of the E-product and we discuss both its properties and the
renormalization ambiguities in the underlying procedure. As last step we extend
the whole analysis to observables admitting derivatives of the field
configurations and we discuss the quantum M{\o}ller operator which is used to
investigate interacting models at a perturbative level.Comment: 35 page
- âŚ