161 research outputs found
Navigating New Trade Routes: The rise of Value chains, and the Challenges for Canadian Trade Policy
In the new paradigm of international trade, Canada needs a trade policy that recognizes both the increasing importance of global value chains and the critical role of Canada-US commercial and regulatory integration in gaining full benefit from their exploitation.border papers, international policy
Mapping grid points onto a square forces an arbitrarily large Lipschitz constant
We prove that the regular square grid of points in the integer
lattice cannot be recovered from an arbitrary -element
subset of via a mapping with prescribed Lipschitz constant
(independent of ). This answers negatively a question of Feige from 2002.
Our resolution of Feige's question takes place largely in a continuous setting
and is based on some new results for Lipschitz mappings falling into two broad
areas of interest, which we study independently. Firstly the present work
contains a detailed investigation of Lipschitz regular mappings on Euclidean
spaces, with emphasis on their bilipschitz decomposability in a sense
comparable to that of the well known result of Jones. Secondly, we build on
work of Burago and Kleiner and McMullen on non-realisable densities. We verify
the existence, and further prevalence, of strongly non-realisable densities
inside spaces of continuous functions.Comment: 60 pages (43 pages of the main part, 13 pages of appendices), 10
figures. This is a revised version according to referees' comments. Our
version of the proof of the theorem about bilipschitz decomposition of
Lipschitz regular mappings was greatly simplified. To appear in GAF
Typical differentiability within an exceptionally small set
We verify the existence of a purely unrectifiable set in which the typical
Lipschitz function has a large set of full differentiability points. The
example arises from a construction, due to Cs\"ornyei, Preiss and Ti\v{s}er, of
a universal differentiability set in which a certain Lipschitz function has
only a purely unrectifiable set of differentiability points.Comment: 33 pages, Appendix 6 pages Accepted for publication in Journal of
Mathematical Analysis and Applications. Some revisions made following the
referees' feedbac
Porosity phenomena of non-expansive, Banach space mappings
For any non-trivial convex and bounded subset of a Banach space, we show
that outside of a -porous subset of the space of non-expansive mappings
, all mappings have the maximal Lipschitz constant one witnessed
locally at typical points of . This extends a result of Bargetz and the
author from separable Banach spaces to all Banach spaces and the proof given is
completely independent. We further establish a fine relationship between the
classes of exceptional sets involved in this statement, captured by the
hierarchy of notions of -porosity.Comment: A few corrections and improvements made. To appear in Israel Journal
of Mathematic
Highly irregular separated nets
In 1998 Burago and Kleiner and (independently) McMullen gave examples of separated nets in Euclidean space which are non-bilipschitz equivalent to the integer lattice. We study weaker notions of equivalence of separated nets and demonstrate that such notions also give rise to distinct equivalence classes. Put differently, we find occurrences of particularly strong divergence of separated nets from the integer lattice. Our approach generalises that of Burago and Kleiner and McMullen which takes place largely in a continuous setting. Existence of irregular separated nets is verified via the existence of non-realisable density functions ρ:[0,1]d→(0,∞). In the present work we obtain stronger types of non-realisable densities
A dichotomy of sets via typical differentiability
We obtain a criterion for an analytic subset of a Euclidean space to contain
points of differentiability of a typical Lipschitz function, namely, that it
cannot be covered by countably many sets, each of which is closed and purely
unrectifiable (has zero length intersection with every curve).
Surprisingly, we establish that any set failing this criterion witnesses the
opposite extreme of typical behaviour: In any such coverable set a typical
Lipschitz function is everywhere severely non-differentiable.Comment: Accepted for publication in Forum of Mathematics Sigma. Some
revisions made according to the referee's repor
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