Dispersive Estimates of Solutions to the Wave Equation with a Potential in Dimensions Two and Three

2000 Mathematics Subject Classification: 35L15, 35B40, 47F05.

Weighted Dispersive Estimates for Solutions of the Schrödinger Equation

2000 Mathematics Subject Classification: 35L15, 35B40, 47F05.

Dispersive estimates for the Schrödinger equation

In this document we explore the issue of $L^1\to L^\infty$ estimates for the solution operator of the linear Schr\"{o}dinger equation, \begin{align*} iu_t-\Delta u+Vu&=0 &u(x,0)=f(x)\in \mathcal S(\R^n). \end{align*} We focus particularly on the five and seven dimensional cases. We prove that the solution operator precomposed with projection onto the absolutely continuous spectrum of $H=-\Delta+V$ satisfies the following estimate $\|e^{itH} P_{ac}(H)\|_{L^1\to L^\infty} \lesssim |t|^{-\frac{n}{2}}$ under certain conditions on the potential $V$. Specifically, we prove the dispersive estimate is satisfied with optimal assumptions on smoothness, that is $V\in C^{\frac{n-3}{2}}(\R^n)$ for $n=5,7$ assuming that zero is regular, $|V(x)|\lesssim \langle x\rangle^{-\beta}$ and $|\nabla^j V(x)|\lesssim \langle x\rangle^{-\alpha}$, $1\leq j\leq \frac{n-3}{2}$ for some $\beta>\frac{3n+5}{2}$ and $\alpha>3,8$ in dimensions five and seven respectively. We also show that for the five dimensional result one only needs that $|V(x)|\lesssim \langle x\rangle^{-4-}$ in addition to the assumptions on the derivative and regularity of the potential. This more than cuts in half the required decay rate in the first chapter. Finally we consider a problem involving the non-linear Schr\"{o}dinger equation. In particular, we consider the following equation that arises in fiber optic communication systems, \begin{align*} iu_t+d(t) u_{xx}+|u|^2 u=0. \end{align*} We can reduce this to a non-linear, non-local eigenvalue equation that describes the so-called dispersion management solitons. We prove that the dispersion management solitons decay exponentially in $x$ and in the Fourier transform of $x$.

Error estimates for a fully discrete spectral scheme for a class of nonlinear, nonlocal dispersive wave equations

We analyze a fully discrete spectral method for the numerical solution of the initial- and periodic boundary-value problem for two nonlinear, nonlocal, dispersive wave equations, the Benjamin–Ono and the Intermediate Long Wave equations. The equations are discretized in space by the standard Fourier–Galerkin spectral method and in time by the explicit leap-frog scheme. For the resulting fully discrete, conditionally stable scheme we prove an L2-error bound of spectral accuracy in space and of second-order accuracy in time.

Joint Antenna Selection and Frequency-Domain Scheduling in OFDMA Systems with Imperfect Estimates from Dual Pilot Training Scheme

Transmit antenna selection (AS) has been adopted in contemporary wideband wireless standards such as Long Term Evolution (LTE). We analyze a comprehensive new model for AS that captures several key features about its operation in wideband orthogonal frequency division multiple access (OFDMA) systems. These include the use of channel-aware frequency-domain scheduling (FDS) in conjunction with AS, the hardware constraint that a user must transmit using the same antenna over all its assigned subcarriers, and the scheduling constraint that the subcarriers assigned to a user must be contiguous. The model also captures the novel dual pilot training scheme that is used in LTE, in which a coarse system bandwidth-wide sounding reference signal is used to acquire relatively noisy channel state information (CSI) for AS and FDS, and a dense narrow-band demodulation reference signal is used to acquire accurate CSI for data demodulation. We analyze the symbol error probability when AS is done in conjunction with the channel-unaware, but fair, round-robin scheduling and with channel-aware greedy FDS. Our results quantify how effective joint AS-FDS is in dispersive environments, the interactions between the above features, and the ability of the user to lower SRS power with minimal performance degradation.

Reduced-complexity iterative equalization for severe time-dispersive MIMO channels

In this paper, a reduced-complexity soft-interference-cancellation minimum mean-square-error.(SIC-MMSE) iterative equalization method for severe time-dispersive multiple-input-multiple-output (MIMO) channels is proposed. To mitigate the severe time dispersiveness of the channel, a single carrier with cyclic prefix is employed, and the equalization is per-formed in the frequency domain. This simplifies the challenging problem of equalization in MIMO channels due to both the intersymbol interference (ISI) and the coantenna interference (CAI). The proposed iterative algorithm works in two stages. The first stage estimates the transmitted frequency-domain symbols using a low-complexity SIC-MMSE equalizer. The second stage converts the estimated frequency-domain symbols in the time domain and finds their means and variances to incorporate in the SIC-MMSE equalizer in the next iteration. Simulation results show the bit-/symbol-error-rate performance of the SIC-MMSE equalizer, with and without coding, for various modulation schemes.

Identification of critical input variables for risk-based cost estimates for road maintenance and rehabilitation

An estimation of costs for maintenance and rehabilitation is subject to variation due to the uncertainties of input parameters. This paper presents the results of an analysis to identify input parameters that affect the prediction of variation in road deterioration. Road data obtained from 1688 km of a national highway located in the tropical northeast of Queensland in Australia were used in the analysis. Data were analysed using a probability-based method, the Monte Carlo simulation technique and HDM-4’s roughness prediction model. The results of the analysis indicated that among the input parameters the variability of pavement strength, rut depth, annual equivalent axle load and initial roughness affected the variability of the predicted roughness. The second part of the paper presents an analysis to assess the variation in cost estimates due to the variability of the overall identified critical input parameters.

Population-based survival estimates for childhood cancer in Australia during the period 1997–2006

Background: This study provides the latest available relative survival data for Australian childhood cancer patients. Methods: Data from the population-based Australian Paediatric Cancer Registry were used to describe relative survival outcomes using the period method for 11 903 children diagnosed with cancer between 1983 and 2006 and prevalent at any time between 1997 and 2006. Results: The overall relative survival was 90.4% after 1 year, 79.5% after 5 years and 74.7% after 20 years. Where information onstage at diagnosis was available (lymphomas, neuroblastoma, renal tumours and rhabdomyosarcomas), survival was significantly poorer for more-advanced stage. Survival was lower among infants compared with other children for those diagnosed with leukaemia, tumours of the central nervous system and renal tumours but higher for neuroblastoma. Recent improvements in overall childhood cancer survival over time are mainly because of improvements among leukaemia patients. Conclusion: The high and improving survival prognosis for children diagnosed with cancer in Australia is consistent with various international estimates. However, a 5-year survival estimate of 79% still means that many children who are diagnosed with cancer will die within 5 years, whereas others have long-term health morbidities and complications associated with their treatments. It is hoped that continued developments in treatment protocols will result in further improvements in survival.

Estimates of crowding in long-term care: Comparing two approaches

Objective: Because studies of crowding in long-term care settings are lacking, the authors sought to: (1) generate initial estimates of crowding in nursing homes and assisted living facilities; and (2) evaluate two operational approaches to its measurement. ----- ----- Background: Reactions to density and proximity are complex. Greater density intensifies people's reaction to a situation in the direction (positive or negative) that they would react if the situation were to occur under less dense conditions. People with dementia are especially reactive to the environment. ----- ----- Methods: Using a cross-sectional correlational design in nursing homes and assisted living facilities involving 185 participants, multiple observations (N = 6,455) of crowding and other environmental variables were made. Crowding, location, and sound were measured three times per observation; ambiance was measured once. Data analyses consisted of descriptive statistics, t-tests, and one-way analysis of variance. ----- ----- Results: Crowding estimates were higher for nursing homes and in dining and activity rooms. Crowding also varied across settings and locations by time of day. Overall, the interaction of location and time affected crowding significantly (N = 5,559, df [47, 511], F = 105.69, p < .0001); effects were greater within location-by-hour than between location-by-hour, but the effect explained slightly less variance in Long-Term Care Crowding Index (LTC-CI) estimates (47.41%) than location alone. Crowding had small, direct, and highly significant correlations with sound and with the engaging subscale for ambiance; a similar, though inverse, correlation was seen with the soothing subscale for ambiance. ----- ----- Conclusions: Crowding fluctuates consistent with routine activities such as meals in long-term care settings. Furthermore, a relationship between crowding and other physical characteristics of the environment was found. The LTC-CI is likely to be more sensitive than simple people counts when seeking to evaluate the effects of crowding on the behavior of elders-particularly those with dementia-in long-term care settings. aging in place.

Decentralised Data Fusion with Parzen Density Estimates

Decentralised sensor networks typically consist of multiple processing nodes supporting one or more sensors. These nodes are interconnected via wireless communication. Practical applications of Decentralised Data Fusion have generally been restricted to using Gaussian based approaches such as the Kalman or Information Filter This paper proposes the use of Parzen window estimates as an alternate representation to perform Decentralised Data Fusion. It is required that the common information between two nodes be removed from any received estimates before local data fusion may occur Otherwise, estimates may become overconfident due to data incest. A closed form approximation to the division of two estimates is described to enable conservative assimilation of incoming information to a node in a decentralised data fusion network. A simple example of tracking a moving particle with Parzen density estimates is shown to demonstrate how this algorithm allows conservative assimilation of network information.