By D. Kinderlehrer, et al.,

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**Extra resources for An Introduction to Variational Inequals and Their Applns**

**Example text**

Those defined through some properties, may be infinite. Having defined sets we can move on to consider relationships between sets. One of the most basic among those is the relationship of set inclusion. If every element of set A belongs also to set B we call A a subset of B. This is denoted as A c B. If the converse relationship of Be A is also true we say that sets A and B are equal, A=B. It is easy to note that every non-empty set is a subset of itself and of a universal set X. The elements of X that satisfy the property of not belonging to A form a complement of A and are denoted as A.

We subdivide [x]=[-3, 3] into three sub-intervals [xJ1=[-3, -1], [x2]=[-I, I] and [x3]=[1,3]. 47] This is a significant improvement on the centred enclosure of the function f calculated on the whole interval [x]. 005]. However, a further improvement is easily attainable by increasing the number of subdivisions of [x]. b4 I Figure 12. Centered enclosure of the function evaluated at first on sub-intervals and taken as a union of component enclosures. The gradient lines defme intervals enclosing the values of the function evaluated in individual sub-intervals.

T:J [x] ............. I I I Figure 7. J,{R2). The characterization of hyperboxes is analogous to that introduced for intervals in R. The lower left-hand comer of a hyperbox is referred to as a lower bound of the vector interval x" and the top right-hand comer is referred to as an upper bound, x+. These can be expressed in terms of bounds on the component intervals x"=[X"h " X 2, •.. , X"]T n X+= [ + X h X+2, ... , X+n]T 35 2 Sets and Intervals However, while in a one-dimensional case the interval was fully characterized by its width and center, in case of hyperboxes these measures do not capture the shape of the vector interval in Rn.

### An Introduction to Variational Inequals and Their Applns by D. Kinderlehrer, et al.,

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