A Formal Foundation for Variational Programming Using the Choice Calculus

In this thesis, we present semantic equivalence rules for an extension of the choice calculus and sound operations for an implementation of variational lists. The choice calculus is a calculus for describing variation and the formula choice calculus is an extension with formulas. We prove semantic equivalence rules for the formula choice calculus. Variational lists are functional data structures for representing and computing with variation in lists using the choice calculus. We prove map and bind operations are sound for an implementation of variational lists. These proofs are written and verified in the language of the Coq proof assistant

Renormalized Volume

We develop a universal distributional calculus for regulated volumes of metrics that are singular along hypersurfaces. When the hypersurface is a conformal infinity we give simple integrated distribution expressions for the divergences and anomaly of the regulated volume functional valid for any choice of regulator. For closed hypersurfaces or conformally compact geometries, methods from a previously developed boundary calculus for conformally compact manifolds can be applied to give explicit holographic formulae for the divergences and anomaly expressed as hypersurface integrals over local quantities (the method also extends to non-closed hypersurfaces). The resulting anomaly does not depend on any particular choice of regulator, while the regulator dependence of the divergences is precisely captured by these formulae. Conformal hypersurface invariants can be studied by demanding that the singular metric obey, smoothly and formally to a suitable order, a Yamabe type problem with boundary data along the conformal infinity. We prove that the volume anomaly for these singular Yamabe solutions is a conformally invariant integral of a local Q-curvature that generalizes the Branson Q-curvature by including data of the embedding. In each dimension this canonically defines a higher dimensional generalization of the Willmore energy/rigid string action. Recently Graham proved that the first variation of the volume anomaly recovers the density obstructing smooth solutions to this singular Yamabe problem; we give a new proof of this result employing our boundary calculus. Physical applications of our results include studies of quantum corrections to entanglement entropies.Comment: 31 pages, LaTeX, 5 figures, anomaly formula generalized to any bulk geometry, improved discussion of hypersurfaces with boundar

Transverse shear warping functions for anisotropic multilayered plates

In this work, transverse shear warping functions for an equivalent single layer plate model are formulated from a variational approach. The part of the strain energy which involves the shear phenomenon is expressed in function of the warping functions and their derivatives. The variational calculus leads to a differential system of equations which warping functions must verify. Solving this system requires the choice of values for the (global) shear strains and their derivatives. A particular choice, which is justified for cross-ply laminates, leads to excellent results. For single layer isotropic and orthotropic plates, an analytical expression of the warping functions is given. They involve hyperbolic trigonometric functions. They differ from the z - 4/3z3 Reddy's formula which has been found to be a limit of present warping functions for isotropic and moderately thick plates. When the h/L and/or the G13/E1 ratios significantly differ from those of isotropic and moderately thick plates, a difference between present warping functions and Reddy's formula can be observed, even for the isotropic single layer plate. Finite element simulations agree perfectly with the present warping functions in these cases. For multilayer cross-ply configurations, the warping functions are determined using a semi-analytical procedure. They have been compared to results of 3D finite element simulations. They are in excellent agreement. For angle-ply laminates, the above process gives warping functions that seem to have relevant shapes, even if the choice for global shear values cannot be justified in this case. No finite element comparison has been presented at this time because it is difficult to propose boundary conditions and prescribed load that permit to isolate the shear phenomenon

Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, \emph{the Muchnik reals}, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists w\,\forall x\,A(x,wx)$ and a \emph{bounding principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists z\,\forall x\,\exists y\,(y\le_{\mathrm{T}}(x,z)\land A(x,y))$ where $x,y,z$ range over Muchnik reals, $w$ ranges over functions from Muchnik reals to Muchnik reals, and $A(x,y)$ is a formula not containing $w$ or $z$. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems, Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an English translation of Muchnik's 1963 paper on Muchnik degrees.Comment: 44 page

Semantics and Proof Theory of the Epsilon Calculus

The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. The application of this undervalued formalism has been hampered by the absence of well-behaved proof systems on the one hand, and accessible presentations of its theory on the other. One significant early result for the original axiomatic proof system for the epsilon-calculus is the first epsilon theorem, for which a proof is sketched. The system itself is discussed, also relative to possible semantic interpretations. The problems facing the development of proof-theoretically well-behaved systems are outlined.Comment: arXiv admin note: substantial text overlap with arXiv:1411.362