8,990 research outputs found
Lipschitz extensions of definable p-adic functions
In this paper, we prove a definable version of Kirszbraun's theorem in a
non-Archimedean setting for definable families of functions in one variable.
More precisely, we prove that every definable function , where and ,
that is -Lipschitz in the first variable, extends to a definable
function that is
-Lipschitz in the first variable.Comment: 11 page
Micromachined Capacitive Long-Range Displacement Sensor for Nano-Positioning of Microactuator systems
This thesis is about a “Micromachined capacitive long-range displacement sensor for nano-positioning of microactuator systems”. Possible applications of such microsystems are found in future probe-based datastorage, scanning probe microscopy, microbiology, optical lens manipulation, microgrippers and microrobots, etc. These applications require positioning with nanometer precision over a long range (ten’s of micrometers) and benefit from further miniaturization and the application of sub-mm sized Micro Electro Mechanical Systems (MEMS). In many cases open-loop operation is not sufficient and a form of system control is required to combine nanometer accuracy with a large dynamic range and to obtain better system performance. In order to make such systems both economically viable as well as compact, on-chip position sensing appears to be a requirement. The aim is therefore, to obtain optimal performance through an integration of sensor and actuator with micromachining fabrication technology without additional micro assembly
Some fragments of second-order logic over the reals for which satisfiability and equivalence are (un)decidable
We consider the Σ1 0-fragment of second-order logic over the vocabulary h+, ×, 0, 1, <, S1, ..., Ski, interpreted over the reals, where the predicate symbols Si are interpreted as semi-algebraic sets. We show that, in this context, satisfiability of formulas is decidable for the first-order ∃ ∗ - quantifier fragment and undecidable for the ∃ ∗∀- and ∀ ∗ -fragments. We also show that for these three fragments the same (un)decidability results hold for containment and equivalence of formulas.Fil: Grimson, Rafael. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Kuijpers, Bart. Hasselt University; Bélgic
Evaluating geometric queries using few arithmetic operations
Let \cp:=(P_1,...,P_s) be a given family of -variate polynomials with
integer coefficients and suppose that the degrees and logarithmic heights of
these polynomials are bounded by and , respectively. Suppose furthermore
that for each the polynomial can be evaluated using
arithmetic operations (additions, subtractions, multiplications and the
constants 0 and 1). Assume that the family \cp is in a suitable sense
\emph{generic}. We construct a database , supported by an algebraic
computation tree, such that for each the query for the signs of
can be answered using h d^{\cO(n^2)} comparisons and
arithmetic operations between real numbers. The arithmetic-geometric tools
developed for the construction of are then employed to exhibit example
classes of systems of polynomial equations in unknowns whose
consistency may be checked using only few arithmetic operations, admitting
however an exponential number of comparisons
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