Blast Those Stuumps!

You can get rid of troublesome stumps around your farm by blasting with dynamite. The coast isn\u27t large, but there are precautions to take

Wavelet solution of variable order pseudodifferential equations

Sobolev spaces H m(x)(I) of variable order 0<m(x)<1 on an interval I⊂ℝ arise as domains of Dirichlet forms for certain quadratic, pure jump Feller processes X t∈ℝ with unbounded, state-dependent intensity of small jumps. For spline wavelets with complementary boundary conditions, we establish multilevel norm equivalences in H m(x)(I) and prove preconditioning and wavelet matrix compression results for the variable order pseudodifferential generators A of X. Sufficient conditions on A to satisfy a Gårding inequality in H m(x)(I) and time-analyticity of the semigroup T t associated with the Feller process X t are established. As application, wavelet-based algorithms of log-linear complexity are obtained for the valuation of contingent claims on pure jump Feller-Lévy processes X t with state-dependent jump intensity by numerical solution of the corresponding Kolmogoroff equation

Cooling a mechanical resonator via coupling to a tunable double quantum dot

We study the cooling of a mechanical resonator (MR) that is capacitively coupled to a double quantum dot (DQD). The MR is cooled by the dynamical backaction induced by the capacitive coupling between the DQD and the MR. The DQD is excited by a microwave field and afterwards a tunneling event results in the decay of the excited state of the DQD. An important advantage of this system is that both the energy level splitting and the decay rate of the DQD can be well tuned by varying the gate voltage. We find that the steady average occupancy, below unity, of the MR can be achieved by changing both the decay rate of the excited state and the detuning between the transition frequency of the DQD and the microwave frequency, in analogy to the laser sideband cooling of an atom or trapped ion in atomic physics. Our results show that the cooling of the MR to the ground state is experimentally implementable.Comment: 10 pages, 5 figure

Probing tiny motions of nanomechanical resonators: classical or quantum mechanical?

We propose a spectroscopic approach to probe tiny vibrations of a nanomechanical resonator (NAMR), which may reveal classical or quantum behavior depending on the decoherence-inducing environment. Our proposal is based on the detection of the voltage-fluctuation spectrum in a superconducting transmission line resonator (TLR), which is {\it indirectly} coupled to the NAMR via a controllable Josephson qubit acting as a quantum transducer. The classical (quantum mechanical) vibrations of the NAMR induce symmetric (asymmetric) Stark shifts of the qubit levels, which can be measured by the voltage fluctuations in the TLR. Thus, the motion of the NAMR, including if it is quantum mechanical or not, could be probed by detecting the voltage-fluctuation spectrum of the TLR.Comment: 4 pages, 3 figures. to appear in Physical Review Letter

Multiresolution kernel matrix algebra

We propose a sparse arithmetic for kernel matrices, enabling efficient scattered data analysis. The compression of kernel matrices by means of samplets yields sparse matrices such that assembly, addition, and multiplication of these matrices can be performed with essentially linear cost. Since the inverse of a kernel matrix is compressible, too, we have also fast access to the inverse kernel matrix by employing exact sparse selected inversion techniques. As a consequence, we can rapidly evaluate series expansions and contour integrals to access, numerically and approximately in a data-sparse format, more complicated matrix functions such as $A^\alpha$ and $\exp(A)$. By exploiting the matrix arithmetic, also efficient Gaussian process learning algorithms for spatial statistics can be realized. Numerical results are presented to quantify and quality our findings