6,106 research outputs found
Interval Arithmetic and Standardization
Interval arithmetic is arithmetic for continuous sets. Floating-point intervals are intervals of real numbers with floating-point bounds.
Operations for intervals can be efficiently implemented. There is an unanimous agreement, how to define the basic operations,
if we exclude division by an interval containing zero. Hence, it should be standardized.
For division by zero, two options are possible, the clean exception free interval arithmetic or the containment arithmetic.
They can be standardized as options.
Elementary functions for intervals can be defined. In some application areas loose evaluation of functions,
i.e. evaluation over an interval which is not completely contained in the function domain, is recommended,
In this case, however, a discontinuity flag has to be set to inform that Brouwer\u27s fixed point theorem is no longer applicable in that case
On the Hausdorff dimension of countable intersections of certain sets of normal numbers
We show that the set of numbers that are -distribution normal but not
simply -ratio normal has full Hausdorff dimension. It is further shown under
some conditions that countable intersections of sets of this form still have
full Hausdorff dimension even though they are not winning sets (in the sense of
W. Schmidt). As a consequence of this, we construct many explicit examples of
numbers that are simultaneously distribution normal but not simply ratio normal
with respect to certain countable families of basic sequences. Additionally, we
prove that some related sets are either winning sets or sets of the first
category.Comment: 12 pages, 1 figur
John-type theorems for generalized arithmetic progressions and iterated sumsets
A classical theorem of Fritz John allows one to describe a convex body, up to
constants, as an ellipsoid. In this article we establish similar descriptions
for generalized (i.e. multidimensional) arithmetic progressions in terms of
proper (i.e. collision-free) generalized arithmetic progressions, in both
torsion-free and torsion settings. We also obtain a similar characterization of
iterated sumsets in arbitrary abelian groups in terms of progressions, thus
strengthening and extending recent results of Szemer\'edi and Vu.Comment: 20 pages, no figures, to appear, Adv. in Math. Some minor changes
thanks to referee repor
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