Pointwise Behavior of the Linearized Boltzmann Equation on Torus

We study the pointwise behavior of the linearized Boltzmann equation on torus for non-smooth initial perturbation. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as part of the long-wave expansion in the spectrum of the Fourier mode for the space variable, the time decay rate of the fluid-like waves depends on the size of the domain. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which are carried by the transport equations and have exponential time decay rate. Moreover, the mixture lemma plays an important role in constructing the kinetic-like waves, we supply a new proof of this lemma to avoid constructing explicit solution of the damped transport equation

Novel dip-pen nanolithography strategies for nanopatterning

Dip-pen nanolithography (DPN) is an atomic force microscopy (AFM)-based\ud lithography technique offering the possibility of fabricating patterns with feature sizes\ud ranging from micrometers to tens of nanometers, utilizing either top-down or bottom-up\ud strategies. Although during its early development stages the serial nature of operation of\ud DPN restricted the patterning efficiency, the successful design and fabrication of AFM\ud probe arrays have increased the throughput of DPN dramatically by enabling generation\ud of patterns in parallel. Several advantages of DPN including (1) ease of generation of\ud arbitrary patterns, (2) application to a variety of ink-substrate combinations, and (3) lower\ud cost relative to other lithographic techniques, have led to many applications in different\ud fields and the establishment of DPN as a popular tool for nanofabrication.\ud This thesis contributes to the continuing development of the DPN technique, mainly\ud focusing on two aspects. The first aspect, presented in Chapters 3-5, is to use DPN to\ud fabricate metal-ion and (bio)-molecular patterns at micrometer or sub-micrometer scale\ud onto monolayer-functionalized solid surfaces with different chemical immobilization\ud strategies. Before performing all DPN experiments, microcontact printing (μCP), a\ud well-known and understood simple and rapid soft lithography technique, was used first to\ud confirm the usefulness of the ink-substrate interactions. The second aspect of interest is to\ud develop an AFM tip surface-modification method for long-term DPN writing, as described\ud in Chapters 6 and 7