Write a $3-4$ page research paper on the topic "Memory Hierarchy for Mobile Devices: Understanding the memory hierarchy in mobile architectures, including on$-$chip caches, external memory, and techniques for reducing memory access energy." Your paper should cover the following aspects:

$1.$ Introduction to memory hierarchy and its importance in mobile architectures.

$2.$ Overview of on$-$chip caches in mobile devices, including their role, organization, and hierarchy levels.

$3.$ Discussion on the significance of external memory and its integration into the memory hierarchy of mobile architectures.

$4.$ Analysis of techniques for reducing memory access energy in mobile computing, such as cache compression and data prefetching.

$5.$ Examination of trade$-$offs involved in designing the size and organization of on$-$chip caches in mobile CPUs.

$6.$ Importance of cache coherence and its impact on the memory hierarchy in mobile devices.

$7.$ Discussion on the influence of memory latency on the performance and energy efficiency of mobile architectures.

$8.$ Challenges associated with memory management in mobile devices, including memory allocation and deallocation strategies.

$9.$ Overview of virtual memory and its role in managing memory in mobile architectures.

$10.$ Future trends and advancements in memory hierarchy design for mobile devices.