A pseudo-norm is a particular case of an S-denied norm since we may have vectorial spaces over some given scalar fields where there are some vectors and scalars that satisfy the third axiom [validation], but others that do not satisfy [invalidation
]; or for all vectors and scalars we may have either [parallel]x + y[parallel] = 5 * [parallel]x[parallel]* [parallel]y[parallel] or [parallel]x + y[parallel] = 6 * [parallel]x[parallel] * [parallel]y[parallel], so invalidation
(since we get [parallel]x + y[parallel] > [parallel]x[parallel] * [parallel]y[parallel]) in two different ways.
A number of broadcast based cache invalidation strategies have been proposed for mobile environments.
Hu and Lee  have proposed a family of invalidation algorithms.
Cao [4, 15] has done some pioneering work to reduce the long query delay associated with basic IR based strategy by introducing the idea of updated invalidation report (UIR).
Thus, a large number of uplink requests are also resulted due to the invalidation of cache contents.
To overcome the demerits of existing IR and UIR strategies, this paper presents a synchronous stateful cache invalidation strategy that reduces the query latency, disconnection overheads, use of wireless channel and conserve the client battery energy.
On Improving the Performance of Cache Invalidation in Mobile Environments.
Cache Algorithms Based on Adaptive Invalidation Reports for Mobile Environments.
Specifically, we will investigate the division of the n overlapping subintervals in the report such that the false invalidations can be minimized.
It is observed that the false invalidations are due to the inaccuracy of the report as the exact time of updates has been replaced by fewer rough timestamps (to save bandwidth).
Let us analyze the relationship between the false invalidations and the reconnect distributions.
This measure tells the percentage of correct invalidations per unit length of bit-sequences.
Keywords: Mobile computing, cache invalidation, wireless, data broadcast, invalidation report, disconnection, failure.
Section 3 reviews works in composing invalidation reports.
The cache invalidation method was proposed [Barb 94] to minimize the data transfer in both directions: the downlink and the uplink.