Semismall perturbations, semi-intrinsic ultracontractivity, and integral representations of nonnegative solutions for parabolic equations

We consider nonnegative solutions of a parabolic equation in a cylinder D \timesI, where $D$ is a noncompact domain of a Riemannian manifold and $I =(0,T)$ with $0 < T \le \infty$ or $I=(-\infty,0)$. Under the assumption [SSP] (i.e., the constant function 1 is a semismall perturbation of the associated elliptic operator on $D$), we establish an integral representation theorem of nonnegative solutions: In the case $I =(0,T)$, any nonnegative solution is represented uniquely by an integral on $(D \times \{0 \}) \cup (\partial_M D \times [0,T))$, where $\partial_M D$ is the Martin boundary of $D$ for the elliptic operator; and in the case $I=(-\infty,0)$, any nonnegative solution is represented uniquely by the sum of an integral on $\partial_M D \times (-\infty,0)$ and a constant multiple of a particular solution. We also show that [SSP] implies the condition [SIU] (i.e., the associated heat kernel is semi-intrinsically ultracontractive).Comment: 35 page

On dynamical bit sequences

Let X^{(k)}(t) = (X_1(t), ..., X_k(t)) denote a k-vector of i.i.d. random variables, each taking the values 1 or 0 with respective probabilities p and 1-p. As a process indexed by non-negative t, $X^{(k)}(t)$ is constructed--following Benjamini, Haggstrom, Peres, and Steif (2003)--so that it is strong Markov with invariant measure ((1-p)\delta_0+p\delta_1)^k. We derive sharp estimates for the probability that ``X_1(t)+...+X_k(t)=k-\ell for some t in F,'' where F \subset [0,1] is nonrandom and compact. We do this in two very different settings: (i) Where \ell is a constant; and (ii) Where \ell=k/2, k is even, and p=q=1/2. We prove that the probability is described by the Kolmogorov capacitance of F for case (i) and Howroyd's 1/2-dimensional box-dimension profiles for case (ii). We also present sample-path consequences, and a connection to capacities that answers a question of Benjamini et. al. (2003)Comment: 25 pages. This a substantial revision of an earlier paper. The material has been reorganized, and Theorem 1.3 is ne

Design and Evaluation of a Virtual Quadrant Receiver for 4-ary Pulse Position Modulation/Optical Code Division Multiple Access (4-ary PPM/O-CDMA)

M-ary pulse position modulation (M-ary PPM) is an alternative to on-off-keying (OOK) that transmits multiple bits as a single symbol occupying a frame of M slots. PPM does not require thresholding as in OOK signaling, instead performing a comparison test among all slots in a frame to make the slot decision. Combining PPM with optical code division multiple access (PPM/O-CDMA) adds the benefit of supporting multiple concurrent, asynchronous bursty PPM users. While the advantages of PPM/O-CDMA are well known, implementing a receiver that performs comparison test can be difficult. This paper describes the design of a novel array receiver for M-ary PPM/O-CDMA (M = 4) where the received signal is mapped onto an xy-plane whose quadrants define the PPM slot decision by means of an associated control law. The receiver does not require buffering or nonlinear operations. In this paper we describe a planar lightwave circuit (PLCs) implementation of the receiver. We give detailed numerical simulations that test the concept and investigate the effects of multi-access interference (MAI) and optical beat interference (OBI) on the slot decisions. These simulations provide guidelines for subsequent experimental measurements that will be described