The Geometric Interior in Real Linear Spaces

We introduce the notions of the geometric interior and the centre of mass for subsets of real linear spaces. We prove a number of theorems concerning these notions which are used in the theory of abstract simplicial complexes.Institute of Informatics, University of Białystok, PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Convex sets and convex combinations. Formalized Mathematics, 11(1):53-58, 2003.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Karol Pąk. Affine independence in vector spaces. Formalized Mathematics, 18(1):87-93, 2010, doi: 10.2478/v10037-010-0012-z.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581-588, 1990.Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313-319, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Holomorphic Quantization of Linear Field Theory in the General Boundary Formulation

We present a rigorous quantization scheme that yields a quantum field theory in general boundary form starting from a linear field theory. Following a geometric quantization approach in the K\"ahler case, state spaces arise as spaces of holomorphic functions on linear spaces of classical solutions in neighborhoods of hypersurfaces. Amplitudes arise as integrals of such functions over spaces of classical solutions in regions of spacetime. We prove the validity of the TQFT-type axioms of the general boundary formulation under reasonable assumptions. We also develop the notions of vacuum and coherent states in this framework. As a first application we quantize evanescent waves in Klein-Gordon theory

A Dynamical Approach to the Perron-Frobenius Theory and Generalized Krein-Rutman Type Theorems

We present a new dynamical approach to the classical Perron-Frobenius theory by using some elementary knowledge on linear ODEs. It is completely self-contained and significantly different from those in the literature. As a result, we develop a complex version of the Perron-Frobenius theory and prove a variety of generalized Krein-Rutman type theorems for real operators. In particular, we establish some new Krein-Rutman type theorems for sectorial operators in a formalism that can be directly applied to elliptic operators, which allow us to reduce significantly the technical PDE arguments involved in the study of the principal eigenvalue problems of these operators.Comment: 40 page