Optimal query complexity for estimating the trace of a matrix

Given an implicit $n\times n$ matrix $A$ with oracle access $x^TA x$ for any $x\in \mathbb{R}^n$, we study the query complexity of randomized algorithms for estimating the trace of the matrix. This problem has many applications in quantum physics, machine learning, and pattern matching. Two metrics are commonly used for evaluating the estimators: i) variance; ii) a high probability multiplicative-approximation guarantee. Almost all the known estimators are of the form $\frac{1}{k}\sum_{i=1}^k x_i^T A x_i$ for $x_i\in \mathbb{R}^n$ being i.i.d. for some special distribution. Our main results are summarized as follows. We give an exact characterization of the minimum variance unbiased estimator in the broad class of linear nonadaptive estimators (which subsumes all the existing known estimators). We also consider the query complexity lower bounds for any (possibly nonlinear and adaptive) estimators: (1) We show that any estimator requires $\Omega(1/\epsilon)$ queries to have a guarantee of variance at most $\epsilon$. (2) We show that any estimator requires $\Omega(\frac{1}{\epsilon^2}\log \frac{1}{\delta})$ queries to achieve a $(1\pm\epsilon)$-multiplicative approximation guarantee with probability at least $1 - \delta$. Both above lower bounds are asymptotically tight. As a corollary, we also resolve a conjecture in the seminal work of Avron and Toledo (Journal of the ACM 2011) regarding the sample complexity of the Gaussian Estimator.Comment: full version of the paper in ICALP 201

A statistical model to describe invariants extracted from a 3-D quadric surface patch and its applications in region-based recognition

A statistical model, describing noise-disturbed invariants extracted from a surface patch of a range image, has been developed and applied to region based pose estimation and classification of 3D quadrics. The Mahalanobis distance, which yields the same results as a Baysian classifier, is used for the classification of the surface patches. The results, compared with the Euclidean distance, appear to be much more reliabl

Micromachined Millimetre-Wave Passive Components at 38 and 77 GHz

A precision micro-fabrication technique has been developed for millimetre-wave components of air-filled three-dimensional structures, such as rectangular coaxial lines or waveguides. The devices are formed by bonding several layers of micromachining defined slices with a thickness of a few hundred micrometres. The slices are thickphotoresist SU8 defined by photolithography, or silicon with a pattern defined by deep reactive ion etching; both are coated with gold by evaporation. The process is simple, and low-cost, as compared with conventional precision metal machining, but yields mm-wave components with good performance. The components are light weight and truly airfilled with no dielectric support. This paper reviews several of these micromachined mm-wave components at 38 and 77 GHz for communications and radar applications

Depairing field, onset temperature and the nature of the transition in cuprates

The depairing (upper critical) field $H_{c2}$ in hole-doped cuprates has been inferred from magnetization curves $M$-$H$ measured by torque magnetometry in fields $H$ up to 45 T. We discuss the implications of the results for the pair binding energy, the Nernst onset temperature, fluctuations and the nature of the Meissner transition at $T_c$.Comment: 4 pages, 5 figs., proc. M2S-HTSC-VIII, Dresden 2006, Physica (in press

High resolution forecast models of water vapour over mountains: comparison of results from the UM and MERIS

Propagation delay due to variable tropospheric water vapor (WV) is one of the most intractable problems for radar interferometry, particularly over mountains. The WV field can be simulated by an atmospheric model, and the difference between the two fields is used to correct the radar interferogram. Here, we report our use of the U.K. Met Office Unified Model in a nested mode to produce high-resolution forecast fields for the 3-km-high Mount Etna volcano. The simulated precipitablewater field is validated against that retrieved from the Medium Resolution Imaging Spectrometer (MERIS) radiometer on the Envisat satellite, which has a resolution of 300 m. Two case studies, one from winter (November 24, 2004) and one from summer (June 25, 2005), show that the mismatch between the model and the MERIS fields (rms = 1.1 and 1.6 mm, respectively) is small. One of the main potential sources of error in the models is the timing of the WV field simulation. We show that long-wavelength upper tropospheric troughs of low WV could be identified in both the model output and Meteosat WV imagery for the November 24, 2004 case and used to choose the best time of model output. © 2007 IEEE

The floor in the interplanetary magnetic field: Estimation on the basis of relative duration of ICME observations in solar wind during 1976-2000

To measure the floor in interplanetary magnetic field and estimate the time- invariant open magnetic flux of Sun, it is necessary to know a part of magnetic field of Sun carried away by CMEs. In contrast with previous papers, we did not use global solar parameters: we identified different large-scale types of solar wind for 1976-2000 interval, obtained a fraction of interplanetary CMEs (ICMEs) and calculated magnitude of interplanetary magnetic field B averaged over 2 Carrington rotations. The floor of magnetic field is estimated as B value at solar cycle minimum when the ICMEs were not observed and it was calculated to be 4,65 \pm 6,0 nT. Obtained value is in a good agreement with previous results.Comment: 10 pages, 2 figures, submitted in GR

Very fast formation of superconducting MgB2/Fe wires with high Jc

In this paper we have investigated the effects of sintering time and temperature on the formation and critical current densities of Fe-clad MgB2 wires. MgB2 wires were fabricated using the powder-in-tube process and sintered for different periods of time at predetermined temperatures. All the samples were examined using XRD, SEM and magnetisation measurements. In contrast to the common practice of sintering for several hours, the present results show that there is no need for prolonged heat treatment in the fabrication of Fe-clad MgB2 wires. A total time in the furnace of several minutes is more than enough to form nearly pure MgB2 with high performance characteristics. The results from Tc, Jc and Hirr show convincingly that the samples which were sintered for 3 minutes above 800 oC are as good as those sintered for longer times. In fact, the Jc field performance for the most rapidly sintered sample is slightly better than for all other samples. Jc of 4.5 times 10 ^5 A/cm2 in zero field and above 10 ^5 A/cm2 in 2T at 15 K has been achieved for the best Fe-clad MgB2 wires. As a result of such a short sintering there is no need for using high purity argon protection and it is possible to carry out the heat treatment in a much less protective atmosphere or in air. These findings substantially simplify the fabrication process, making it possible to have a continuous process for fabrication and reducing the costs for large-scale production of MgB2 wires.Comment: 15 pages, one table, 9 figures, submitted to Physica C on June 8, 200

Solitons in systems of coupled scalar fields

We present a method to obtain soliton solutions to relativistic system of coupled scalar fields. This is done by examining the energy associated to static field configurations. In this case we derive a set of first-order differential equations that solve the equations of motion when the energy saturates its lower bound. To illustrate the general results, we investigate some systems described by polynomial interactions in the coupled fields.Comment: RevTex4, 5 page

The role of the Treaty of Waitangi in contemporary public law : does the Treaty have to be incorporated into municipal law to be of effect?

Recent work on Al-Cu-Mg based alloys with Cu:Mg atomic ratio close to unity is reviewed to clarify the mechanisms for age hardening. During the first stage of hardening a substantial exothermic heat evolution occurs whilst the microstructural change involves the formation of initially Cu-rich / Mg-rich clusters and later Cu-Mg co-clusters. The data show that the first stage of the age hardening is due to the formation of Cu-Mg co-clusters. The combined experimental methods show the second stage hardening is dominated by formation of S phase, which forms a dense precipitation at the peak hardness stage, whilst no significant amounts of other phases or zones are detected. S phase strengthens the alloy predominantly through the Orowan looping mechanism. These findings are incorporated into a multi-phase, multi-mechanism model for yield strength of Al-Cu-Mg based alloys