Frames and Phase Retrieval

Phase retrieval tackles the problem of recovering a signal after loss of phase. The phase problem shows up in many different settings such as X-ray crystallography, speech recognition, quantum information theory, and coherent diffraction imaging. In this dissertation we present some results relating to three topics on phase retrieval. Chapters 1 and 2 contain the relevant background materials. In chapter 3, we introduce the notion of exact phase-retrievable frames as a way of measuring a frame\u27s redundancy with respect to its phase retrieval property. We show that, in the d-dimensional real Hilbert space case, exact phase-retrievable frames can be of any lengths between 2d - 1 and d(d + 1)=2, inclusive. The complex Hilbert space case remains open. In chapter 4, we investigate phase-retrievability by studying maximal phase-retrievable subspaces with respect to a given frame. These maximal PR-subspaces can have different dimensions. We are able to identify the ones with the largest dimension and this can be considered as a generalization of the characterization of real phase-retrievable frames. In the basis case, we prove that if M is a k-dimensional PR-subspace then |supp(x)| ≥ k for every nonzero vector x 2 M. Moreover, if 1 ≤ k \u3c [(d + 1)=2], then a k-dimensional PR-subspace is maximal if and only if there exists a vector x ϵ M such that |supp(x)| = k|. Chapter 5 is devoted to investigating phase-retrievable operator-valued frames. We obtain some characterizations of phase-retrievable frames for general operator systems acting on both finite and infinite dimensional Hilbert spaces; thus generalizing known results for vector-valued frames, fusion frames, and frames of Hermitian matrices. Finally, in Chapter 6, we consider the problem of characterizing projective representations that admit frame vectors with the maximal span property, a property that allows for an algebraic recovering of the phase-retrieval problem. We prove that every irreducible projective representation of a finite abelian group admits a frame vector with the maximal span property. All such vectors can be explicitly characterized. These generalize some of the recent results about phase-retrieval with Gabor (or STFT) measurements

Tevatron Combination of Single-Top-Quark Cross Sections and Determination of the Magnitude of the Cabibbo-Kobayashi-Maskawa Matrix Element Vtb\bf V_{tb}

We present the final combination of CDF and D0 measurements of cross sections for single-top-quark production in proton-antiproton collisions at a center-of-mass energy of 1.96 TeV. The data correspond to total integrated luminosities of up to 9.7 fb$^{−1}$ per experiment. The t-channel cross section is measured to be σ$_t$=2.25$_{-0.31}^{+0.29}$ pb. We also present the combinations of the two-dimensional measurements of the s- vs t-channel cross section. In addition, we give the combination of the s+t channel cross section measurement resulting in σ$_{s+t}$=3.30$_{-0.40}^{+0.52}$ pb, without assuming the standard model value for the ratio σ$_s$/σ$_t$. The resulting value of the magnitude of the top-to-bottom quark coupling is |V$_{tb}$|=1.02$_{-0.05}^{+0.06}$, corresponding to |V$_{tb}$|>0.92 at the 95% C.L

Combined Forward-Backward Asymmetry Measurements in Top-Antitop Quark Production at the Tevatron

International audienceThe CDF and D0 experiments at the Fermilab Tevatron have measured the asymmetry between yields of forward- and backward-produced top and antitop quarks based on their rapidity difference and the asymmetry between their decay leptons. These measurements use the full data sets collected in proton-antiproton collisions at a center-of-mass energy of s=1.96  TeV. We report the results of combinations of the inclusive asymmetries and their differential dependencies on relevant kinematic quantities. The combined inclusive asymmetry is AFBtt¯=0.128±0.025. The combined inclusive and differential asymmetries are consistent with recent standard model predictions

Tevatron Run II combination of the effective leptonic electroweak mixing angle

International audienceDrell-Yan lepton pairs produced in the process pp¯→ℓ+ℓ-+X through an intermediate γ*/Z boson have an asymmetry in their angular distribution related to the spontaneous symmetry breaking of the electroweak force and the associated mixing of its neutral gauge bosons. The CDF and D0 experiments have measured the effective-leptonic electroweak mixing parameter sin2θefflept using electron and muon pairs selected from the full Tevatron proton-antiproton data sets collected in 2001-2011, corresponding to 9–10  fb-1 of integrated luminosity. The combination of these measurements yields the most precise result from hadron colliders, sin2θefflept=0.23148±0.00033. This result is consistent with, and approaches in precision, the best measurements from electron-positron colliders. The standard model inference of the on-shell electroweak mixing parameter sin2θW, or equivalently the W-boson mass MW, using the zfitter software package yields sin2θW=0.22324±0.00033 or equivalently, MW=80.367±0.017  GeV/c2