Quasicontinuous functions, domains, and extended calculus
One of the aims of domain theory is the construction of an embedding of a given structure or data type as the maximal or “ideal” elements of an enveloping domain of “approximations,” sometimes called a domain environment. Typically the goal is to provide a computational model or framework for recursive and algorithmic reasoning about the original structure. In this paper we consider the function space of (natural equivalence classes of) quasicontinuous functions from a locally compact space X into L, an n-fold product of the extended reals [−1,1] (more generally, into a bicontinuous lattice). We show that the domain of all “approximate maps” that assign to each point of X an order interval of L is a domain environment for the quasicontinuous function space. We rely upon the theory of domain environments to introduce an interesting and useful function space topology on the quasicontinuous function space. We then apply this machinery to define an extended differential calculus in the quasicontinuous function space, and draw connections with viscosity solutions of Hamiltonian equations. The theory depends heavily on topological properties of quasicontinuous functions that have been recently uncovered that involve dense sets of points of continuity and sections of closed relations and USCO maps. These and other basic results about quasicontinuous functions are surveyed and presented in the early sections.
M. Bardi, M. G. Crandall, L. C. Evans, H. M. Soner and P. E. Souganidis, Viscosity So lutions and Applications, Lectures Notes in Mathematics 1660, Springer-Verlag, Berlin, Heidelberg, 1997
J. Borsík, Products of simply continuous and quasicontinuous functions, Math. Slovaca 45 (1995), 445–452.
J. Borsík, Maxima and minima of simply continuous and quasicontinuous functions, Math. Slovaca 46 (1996), 261–268.
J. Cao and W. Moors, Quasicontinuous selections of upper continuous set-valued mappings, Real Anal. Exchange 31 (2005), 63–72.
B. Cascales and L. Oncina, Compactoid filters and USCO maps, J. Math. Anal Appl. 283 (2003), 826–845. http://dx.doi.org/10.1016/S0022-247X(03)00280-4
R. Cazacu, Quasicontinuous derivatives and viscosity functions, Dissertation, Louisiana State University, 2005.
A. Crannell, M. Frantz and M. LeMasurier, Closed relations and equivalence classes of quasicontinuous functions, Real Anal. Exchange 31 (2006), 409–424.
A. Edalat and A. Lieutier, Domain Theory and Differential Calculus (Functions of one variable), Mathematical Structures in Computer Science 14 (2004), 771–802. http://dx.doi.org/10.1017/S0960129504004359
A. Edalat, A. Lieutier, and D. Pattinson, A computational model for multi-variable differential calculus, Proceedings of FOSSACS 2005, 26 pages.
G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M.W. Mislove and D.S. Scott, Continuous Lattices and Domains, Cambridge University Press, 2003. http://dx.doi.org/10.1017/CBO9780511542725
S. Kempisty, Sur les fonctions quasicontinues, Fund. Math. 19 (1932), 184–197.
J. Lawson, Encounters between topology and domain theory, Domains and Processes, Kluwer Academic Publishers, Netherlands, 2001, 1–32. http://dx.doi.org/10.1007/978-94-010-0654-5_1
K. Martin, The informatic derivative at a compact element, Proc. FoSSaCS02, Springer LNCS 2303 (2002), 310–325.
K. Martin and J. Ouaknine, Informatic vs. classical differentation on the real line, Electronic Notes in Theor. Comp. Sci. 73 (2003), 8 pages. URL: http://www.elsevier.nl/locate/entcs/volume73.html
M. Matejdes, Sur les sélecteurs des multifonction, Math. Slovaca 37 (1987), 111–124.
W. Miller and E. Akin, Invariant measure for set-valued dynamical systems, Trans. Amer. Math. Soc. 351:3 (1999) 1203-1225. http://dx.doi.org/10.1090/S0002-9947-99-02424-1
T. Neubrunn, Quasi-continuity, Real Anal. Exchange 14 (1988/89), 259–306.
H.L. Royden, Real Analysis, Macmillan, New York, 1965.
S. Samborski, A new function space and extension of partial differential operators in it, preprint.
S. Samborski, Expansions of differential operators and nonsmooth solutions of differential equations, Cybern. Syst. Anal. 38 (3) (2002), 453–466. http://dx.doi.org/10.1023/A:1020325013640
Metrics powered by PLOS ALM
This journal is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Universitat Politècnica de València
e-ISSN: 1989-4147 https://doi.org/10.4995/agt