1,291 research outputs found
Smooth Parametrizations in Dynamics, Analysis, Diophantine and Computational Geometry
Smooth parametrization consists in a subdivision of the mathematical objects
under consideration into simple pieces, and then parametric representation of
each piece, while keeping control of high order derivatives. The main goal of
the present paper is to provide a short overview of some results and open
problems on smooth parametrization and its applications in several apparently
rather separated domains: Smooth Dynamics, Diophantine Geometry, Approximation
Theory, and Computational Geometry.
The structure of the results, open problems, and conjectures in each of these
domains shows in many cases a remarkable similarity, which we try to stress.
Sometimes this similarity can be easily explained, sometimes the reasons remain
somewhat obscure, and it motivates some natural questions discussed in the
paper. We present also some new results, stressing interconnection between
various types and various applications of smooth parametrization
On the complexity of nonlinear mixed-integer optimization
This is a survey on the computational complexity of nonlinear mixed-integer
optimization. It highlights a selection of important topics, ranging from
incomputability results that arise from number theory and logic, to recently
obtained fully polynomial time approximation schemes in fixed dimension, and to
strongly polynomial-time algorithms for special cases.Comment: 26 pages, 5 figures; to appear in: Mixed-Integer Nonlinear
Optimization, IMA Volumes, Springer-Verla
Polynomial Size Analysis of First-Order Shapely Functions
We present a size-aware type system for first-order shapely function
definitions. Here, a function definition is called shapely when the size of the
result is determined exactly by a polynomial in the sizes of the arguments.
Examples of shapely function definitions may be implementations of matrix
multiplication and the Cartesian product of two lists. The type system is
proved to be sound w.r.t. the operational semantics of the language. The type
checking problem is shown to be undecidable in general. We define a natural
syntactic restriction such that the type checking becomes decidable, even
though size polynomials are not necessarily linear or monotonic. Furthermore,
we have shown that the type-inference problem is at least semi-decidable (under
this restriction). We have implemented a procedure that combines run-time
testing and type-checking to automatically obtain size dependencies. It
terminates on total typable function definitions.Comment: 35 pages, 1 figur
Purely exponential parametrizations and their group-theoretic applications
This paper is mainly motivated by the analysis of the so-called Bounded
Generation property (BG) of linear groups (in characteristic ), which is
known to admit far-reaching group-theoretic implications.
We achieve complete answers to certain longstanding open questions about
Bounded Generation (sharpening considerably some earlier results). For
instance, we prove that linear groups boundedly generated by semi-simple
elements are necessarily virtually abelian. This is obtained as a corollary of
sparseness of subsets which are likewise generated.
In the paper in fact we go further, framing (BG) in the more general context
of (Purely) Exponential Parametrizations (PEP) for subsets of affine spaces, a
concept which unifies different issues. Using deep tools from Diophantine
Geometry (including the Subspace Theorem), we systematically develop a theory
showing in particular that for a (PEP) set over a number field, the asymptotic
distribution of its points of Height at most is always ,
with certain constants and . (This shape fits
with a well-known viewpoint first put forward by Manin.
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