As the semiconductor process nodes advance to 28nm and below and three-dimensional (3D) silicon integration technology is emerging, power delivery is becoming an ever -greater challenge in VLSI design. Detailed analyses of power distribution networks (PDN) must be performed in order to ensure robust power delivery. Different aspects of analysis problems with current and future VLSI designs and 3D ICs are addressed in this dissertation. We review the traditional frequency-domain PDN analysis methodology and investigate the relationship between the frequency- domain PDN impedance and time-domain PDN noise. We demonstrate that the traditional frequency-domain analysis methodology has limitations. Due to the limitation of frequency-domain analysis methodology, we point out the importance of time-domain analysis which lead to our research work worst-case PDN noise prediction. We also propose a simulation flow for large-scale PDN using Discrete Fourier Transform (DFT). This flow utilizes the characterizations of PDNs and adaptively choose the simulation points based on different frequency ranges of PDNs. With such an adaptive sampling technique, we are able to simulate large-scale PDN models and achieve fairly results in a reasonable simulation time. Our simulation flow also naturally fits into parallel computation techniques, which further enhances the simulation efficiency. Finally, with our analysis and simulation tools, we explore the PDN characteristics of 3D ICs. We propose lumped and distributed models for 3D PDNs, and analyze 3D PDNs systematically in both frequency domain and time domain. Unique noise behaviors of 3D PDNs are discovered with our modeling analysis flow. Different design space of 3D PDNs are also studied and design guidelines based on the studies are provided

In this note, we study Schwarz's conjecture on application of Q-algebras to strict quantization. We prove that in the case of a torus with a constant Poisson structure, Schwarz's formalism gives the same star product as Rieffel \cite{rif:quantization}. We construct twisted Fock modules as examples of quantization dg-modules in the case of a compact K\"ahler manifold. In particular, we relate this construction on $\complex\mathbb{P}^1$ to representations of a fuzzy sphere.

The dissertation investigates two major problems in making a remote collaborative recording: echo cancellation during the recording, and signal separation during post processing. A prototype system is conceived and simulated to demonstrate the validity of the design in separated steps

Let X and X' be closed subschemes of an algebraic torus T over a non-archimedean field. We prove the rational equivalence as tropical cycles, in the sense of Henning Meyer's graduate thesis, between the tropicalization of the intersection product of X and X' and the stable intersection of trop(X) and trop(X'), when restricted to (the inverse image under the tropicalization map of) a connected component C of the intersection of trop(X) and trop(X'). This requires possibly passing to a (partial) compactification of T with respect to a suitable fan. We define the compactified stable intersection in a toric tropical variety, and check that this definition is compatible with the intersection product in loc.cit.. As a result we get a numerical equivalence (after a compactification and restricting to C) between the intersection product of X and X' and the stable intersection of trop(X) and trop(X') via the compactified stable intersection. In particular, when X and X' have complementary codimensions, this equivalence generalizes the work of Osserman and Rabinoff, in the sense that the intersection of X and X' is allowed to be of positive dimension. Moreover, if the intersection of the closures of X and X' has finitely many points which tropicalize to the closure of C, we prove a similar equation as in Theorem 6.4 of the paper of Osserman and Rabinoff when the ambient space is a reduced closed subscheme of T (instead of T itself).

In this note, we will show one example of hamiltonian Lie algebra action which has no invariant star product.

The ever-growing Internet, among other things, provides abundant evidence that we are living in a world of interaction. It is thus of great importance to understand the benefits of interaction in our lives. In this thesis, we investigate the role of interaction in information theory and statistics via three concrete problems: distributed inference, point-to-point channel communication, and communication over networks. First, we consider a classical hypothesis testing problem in a distributed setting, where communication constraints are present. More specifically, two distributed agents, having only partial access to some random data set, are required to perform a hypothesis test regarding the joint data distribution by communicating their observations with each other. The goal is to characterize the optimal tradeoff between the testing performance and the communication budget. We formulate an interactive version of this problem, where the two agents are allowed to communicate in multiple rounds before making a decision. Interestingly, the testing performance can be strictly improved given the same communication budget. Moreover, we study a sequential version of the interactive hypothesis problem which further improves the testing performance. Second, we investigate the role of interaction in the reliability of communication by studying the optimal coding over the Gaussian channel with noisy Gaussian feedback. While it is well known that the reliability of communication can be strictly improved through noiseless feedback, the theoretical understanding of the benefits of noisy feedback is yet far from being complete. We propose two coding schemes that enable strict improvement of the reliability of communication when the noise power in feedback channel is smaller than a certain threshold. Finally, we study the impact of interaction on relay networks. In particular, we focus on a network communication problem, in which one nodes wishes to broadcast a common message to all the other nodes in the network. Typically, relaying schemes are non-interactive, which can be improved by two-round interaction schemes. We investigate this problem beyond the two-round case and demonstrate via a simple example that infinite rounds of interaction can further improve upon finite rounds interactive schemes