46,497 research outputs found
Non-Boolean almost perfect nonlinear functions on non-Abelian groups
The purpose of this paper is to present the extended definitions and
characterizations of the classical notions of APN and maximum nonlinear Boolean
functions to deal with the case of mappings from a finite group K to another
one N with the possibility that one or both groups are non-Abelian.Comment: 17 page
Inhomogeneous extreme forms
G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two
reduction theories for lattices, well-suited for sphere packing and covering
problems. In his first memoir a characterization of locally most economic
packings is given, but a corresponding result for coverings has been missing.
In this paper we bridge the two classical memoirs.
By looking at the covering problem from a different perspective, we discover
the missing analogue. Instead of trying to find lattices giving economical
coverings we consider lattices giving, at least locally, very uneconomical
ones. We classify local covering maxima up to dimension 6 and prove their
existence in all dimensions beyond.
New phenomena arise: Many highly symmetric lattices turn out to give
uneconomical coverings; the covering density function is not a topological
Morse function. Both phenomena are in sharp contrast to the packing problem.Comment: 22 pages, revision based on suggestions by referee, accepted in
Annales de l'Institut Fourie
A Characterization of Scale Invariant Responses in Enzymatic Networks
An ubiquitous property of biological sensory systems is adaptation: a step
increase in stimulus triggers an initial change in a biochemical or
physiological response, followed by a more gradual relaxation toward a basal,
pre-stimulus level. Adaptation helps maintain essential variables within
acceptable bounds and allows organisms to readjust themselves to an optimum and
non-saturating sensitivity range when faced with a prolonged change in their
environment. Recently, it was shown theoretically and experimentally that many
adapting systems, both at the organism and single-cell level, enjoy a
remarkable additional feature: scale invariance, meaning that the initial,
transient behavior remains (approximately) the same even when the background
signal level is scaled. In this work, we set out to investigate under what
conditions a broadly used model of biochemical enzymatic networks will exhibit
scale-invariant behavior. An exhaustive computational study led us to discover
a new property of surprising simplicity and generality, uniform linearizations
with fast output (ULFO), whose validity we show is both necessary and
sufficient for scale invariance of enzymatic networks. Based on this study, we
go on to develop a mathematical explanation of how ULFO results in scale
invariance. Our work provides a surprisingly consistent, simple, and general
framework for understanding this phenomenon, and results in concrete
experimental predictions
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