Matrix probing and its conditioning

When a matrix A with n columns is known to be well approximated by a linear combination of basis matrices B_1,..., B_p, we can apply A to a random vector and solve a linear system to recover this linear combination. The same technique can be used to recover an approximation to A^-1. A basic question is whether this linear system is invertible and well-conditioned. In this paper, we show that if the Gram matrix of the B_j's is sufficiently well-conditioned and each B_j has a high numerical rank, then n {proportional} p log^2 n will ensure that the linear system is well-conditioned with high probability. Our main application is probing linear operators with smooth pseudodifferential symbols such as the wave equation Hessian in seismic imaging. We demonstrate numerically that matrix probing can also produce good preconditioners for inverting elliptic operators in variable media

Riesz transforms on generalized Heisenberg groups and Riesz transforms

Let 1 < q < \infty. We prove that the Riesz transforms $R_{k}=X_{k} L^{-\frac{1}{2}}$ on a generalized Heisenberg group $G$ satisfy $\left\|\left(\sum_{k=1}^{K}\left| R_{k}(f)\right| ^{2}\right)^{\frac{1}{2}}\right\| _{L^{q}(G)}\leq C(q,J)\left\| f\right\| _{L^{q}(G)}$ where $K$, $J$ are respectively the dimensions of the first and second layer of the Lie algebra of $G$. We prove similar inequalities on Schatten spaces $S^{q}(H)$, with dimension free constants, for Riesz transforms associated to commuting inner $*$-derivations $D_{k}$ and a suitable substitute of the square function. An example is given by the derivations associated to $n$ commuting pairs of operators $(P_{j},Q_{j})$ on a Hilbert space $H$ satisfying the canonical commutation relations [P$_{j},Q_{j}]=iI_{H}$

Structured Random Matrices

Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact or approximate symmetries, such as matrices with i.i.d. entries, for which precise analytic results and limit theorems are available. Much less well understood are matrices that are endowed with an arbitrary structure, such as sparse Wigner matrices or matrices whose entries possess a given variance pattern. The challenge in investigating such structured random matrices is to understand how the given structure of the matrix is reflected in its spectral properties. This chapter reviews a number of recent results, methods, and open problems in this direction, with a particular emphasis on sharp spectral norm inequalities for Gaussian random matrices.Comment: 46 pages; to appear in IMA Volume "Discrete Structures: Analysis and Applications" (Springer

Quasiparticle Andreev scattering in the ν=1/3\nu=1/3 fractional quantum Hall regime

The scattering of exotic quasiparticles may follow different rules than electrons. In the fractional quantum Hall regime, a quantum point contact (QPC) provides a source of quasiparticles with field effect selectable charges and statistics, which can be scattered on an 'analyzer' QPC to investigate these rules. Remarkably, for incident quasiparticles dissimilar to those naturally transmitted across the analyzer, electrical conduction conserves neither the nature nor the number of the quasiparticles. In contrast with standard elastic scattering, theory predicts the emergence of a mechanism akin to the Andreev reflection at a normal-superconductor interface. Here, we observe the predicted Andreev-like reflection of an $e/3$ quasiparticle into a $-2e/3$ hole accompanied by the transmission of an $e$ quasielectron. Combining shot noise and cross-correlation measurements, we independently determine the charge of the different particles and ascertain the coincidence of quasielectron and fractional hole. The present work advances our understanding on the unconventional behavior of fractional quasiparticles, with implications toward the generation of novel quasi-particles/holes and non-local entanglements

Observing the universal screening of a Kondo impurity

The Kondo effect, deriving from a local magnetic impurity mediating electron-electron interactions, constitutes a flourishing basis for understanding a large variety of intricate many-body problems. Its experimental implementation in tunable circuits has made possible important advances through well-controlled investigations. However, these have mostly concerned transport properties, whereas thermodynamic observations - notably the fundamental measurement of the spin of the Kondo impurity - remain elusive in test-bed circuits. Here, with a novel combination of a "charge" Kondo circuit with a charge sensor, we directly observe the state of the impurity and its progressive screening. We establish the universal renormalization flow from a single free spin to a screened singlet, the associated reduction in the magnetization, and the relationship between scaling Kondo temperature and microscopic parameters. In our device, a Kondo pseudospin is realized by two degenerate charge states of a metallic island, which we measure with a non-invasive, capacitively coupled charge sensor. Such pseudospin probe of an engineered Kondo system opens the way to the thermodynamic investigation of many exotic quantum states, including the clear observation of Majorana zero modes through their fractional entropy

User-friendly tail bounds for sums of random matrices

This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales. In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. The matrix inequalities promise the same diversity of application, ease of use, and strength of conclusion that have made the scalar inequalities so valuable.Comment: Current paper is the version of record. The material on Freedman's inequality has been moved to a separate note; other martingale bounds are described in Caltech ACM Report 2011-0

Systematic stratospheric observations on the Antarctic continent at Dumont d'Urville

Results of different routine measurements performed in Dumont d'Urville (66 deg S, 140 deg E) since 1988 are presented. They include the seasonal variation of total ozone and NO2 as measured by a SAOZ UV-Visible spectrometer, Polar Stratospheric Cloud observations by a backscatter lidar and more recently, vertical ozone profiles by ECC sondes and ozone and aerosols stratospheric profiles by a DIAL lidar. The particular results of 1991 in relation with the volcanic events of Mount Pinatubo and Mount Hudson, and the position of the polar vortex over Dumont d'Urville are discussed

p53-dependent control of transactivation of the Pen2 promoter by presenilins

The senile plaques found in the brains of patients with Alzheimer's disease are mainly due to the accumulation of amyloid β-peptides (Aβ) that are liberated by γ-secretase, a high molecular weight complex including presenilins, PEN-2, APH-1 and nicastrin. The depletion of each of these proteins disrupts the complex assembly into a functional protease. Here, we describe another level of regulation of this multimeric protease. The depletion of both presenilins drastically reduces Pen2 mRNA levels and its promoter transactivation. Furthermore, overexpression of presenilin-1 lowers Pen2 promoter transactivation, a phenotype abolished by a double mutation known to prevent presenilin-dependent γ-secretase activity. PEN-2 expression is decreased by depletion of β-amyloid precursor protein (APP) and increased by the APP intracellular domain (AICD). We show that AICD and APP complement for Pen2 mRNA levels in APP/APLP1-2 knockout fibroblasts. Interestingly, overexpression of presenilin-2 greatly increases Pen2 promoter transactivation. The opposite effect triggered by both presenilins was reminiscent of our previous study, which showed that these two proteins elicit antagonistic effects on p53. Therefore, we examined the contribution of p53 on Pen2 transcription. Pen2 promoter transactivation, and Pen2 mRNA and protein levels were drastically reduced in p53–/– fibroblasts. Furthermore, PEN-2 expression could be rescued by p53 complementation in p53- and APP-deficient cells. Interestingly, PEN-2 expression was also reduced in p53-deficient mouse brain. Overall, our study describes a p53-dependent regulation of PEN-2 expression by other members of the γ-secretase complex, namely presenilins

Cellular Prion Protein Expression Is Not Regulated by the Alzheimer's Amyloid Precursor Protein Intracellular Domain

There is increasing evidence of molecular and cellular links between Alzheimer's disease (AD) and prion diseases. The cellular prion protein, PrPC, modulates the post-translational processing of the AD amyloid precursor protein (APP), through its inhibition of the β-secretase BACE1, and oligomers of amyloid-β bind to PrPC which may mediate amyloid-β neurotoxicity. In addition, the APP intracellular domain (AICD), which acts as a transcriptional regulator, has been reported to control the expression of PrPC. Through the use of transgenic mice, cell culture models and manipulation of APP expression and processing, this study aimed to clarify the role of AICD in regulating PrPC. Over-expression of the three major isoforms of human APP (APP695, APP751 and APP770) in cultured neuronal and non-neuronal cells had no effect on the level of endogenous PrPC. Furthermore, analysis of brain tissue from transgenic mice over-expressing either wild type or familial AD associated mutant human APP revealed unaltered PrPC levels. Knockdown of endogenous APP expression in cells by siRNA or inhibition of γ-secretase activity also had no effect on PrPC levels. Overall, we did not detect any significant difference in the expression of PrPC in any of the cell or animal-based paradigms considered, indicating that the control of cellular PrPC levels by AICD is not as straightforward as previously suggested