2,684 research outputs found
Formalization of Complex Vectors in Higher-Order Logic
Complex vector analysis is widely used to analyze continuous systems in many
disciplines, including physics and engineering. In this paper, we present a
higher-order-logic formalization of the complex vector space to facilitate
conducting this analysis within the sound core of a theorem prover: HOL Light.
Our definition of complex vector builds upon the definitions of complex numbers
and real vectors. This extension allows us to extensively benefit from the
already verified theorems based on complex analysis and real vector analysis.
To show the practical usefulness of our library we adopt it to formalize
electromagnetic fields and to prove the law of reflection for the planar waves.Comment: 15 pages, 1 figur
Proof-checking Euclid
We used computer proof-checking methods to verify the correctness of our
proofs of the propositions in Euclid Book I. We used axioms as close as
possible to those of Euclid, in a language closely related to that used in
Tarski's formal geometry. We used proofs as close as possible to those given by
Euclid, but filling Euclid's gaps and correcting errors. Euclid Book I has 48
propositions, we proved 235 theorems. The extras were partly "Book Zero",
preliminaries of a very fundamental nature, partly propositions that Euclid
omitted but were used implicitly, partly advanced theorems that we found
necessary to fill Euclid's gaps, and partly just variants of Euclid's
propositions. We wrote these proofs in a simple fragment of first-order logic
corresponding to Euclid's logic, debugged them using a custom software tool,
and then checked them in the well-known and trusted proof checkers HOL Light
and Coq.Comment: 53 page
Covariantly constant forms on torsionful geometries from world-sheet and spacetime perspectives
The symmetries of two-dimensional supersymmetric sigma models on target
spaces with covariantly constant forms associated to special holonomy groups
are analysed. It is shown that each pair of such forms gives rise to a new one,
called a Nijenhuis form, and that there may be further reductions of the
structure group. In many cases of interest there are also covariantly constant
one-forms which also give rise to symmetries. These geometries are of interest
in the context of heterotic supergravity solutions and the associated
reductions are studied from a spacetime point of view via the Killing spinor
equations.Comment: 33 pages, minor modifications, version published in JHE
A tour on Hermitian symmetric manifolds
Hermitian symmetric manifolds are Hermitian manifolds which are homogeneous
and such that every point has a symmetry preserving the Hermitian structure.
The aim of these notes is to present an introduction to this important class of
manifolds, trying to survey the several different perspectives from which
Hermitian symmetric manifolds can be studied.Comment: 56 pages, expanded version. Written for the Proceedings of the
CIME-CIRM summer course "Combinatorial Algebraic Geometry". Comments are
still welcome
Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations
We present a formal tool for verification of multivariate nonlinear
inequalities. Our verification method is based on interval arithmetic with
Taylor approximations. Our tool is implemented in the HOL Light proof assistant
and it is capable to verify multivariate nonlinear polynomial and
non-polynomial inequalities on rectangular domains. One of the main features of
our work is an efficient implementation of the verification procedure which can
prove non-trivial high-dimensional inequalities in several seconds. We
developed the verification tool as a part of the Flyspeck project (a formal
proof of the Kepler conjecture). The Flyspeck project includes about 1000
nonlinear inequalities. We successfully tested our method on more than 100
Flyspeck inequalities and estimated that the formal verification procedure is
about 3000 times slower than an informal verification method implemented in
C++. We also describe future work and prospective optimizations for our method.Comment: 15 page
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