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A necessary and sufficient condition is given to ensure the boundedness of Fourier–Haar multiplier operators from L1([0,1],X) to L1([0,1],Y), where X is an arbitrary finite dimensional Banach space and Y is an arbitrary Banach space. The Fourier–Haar multiplier sequences come not from R, as in the classical case, but from the space of bounded operators from the Banach space X to the Banach space Y. Moreover, it is shown that this condition characterises the finite dimensionality of the Banach space X.

Mesons comprising a beauty quark and strange quark can oscillate between particle ( B s 0 ) and antiparticle ( B ¯ s 0 ) flavour eigenstates, with a frequency given by the mass difference between heavy and light mass eigenstates, Δ m s . Here we present a measurement of Δ m s using B s 0 → D s − π + decays produced in proton–proton collisions collected with the LHCb detector at the Large Hadron Collider. The oscillation frequency is found to be Δ m s  = 17.7683 ± 0.0051 ± 0.0032 ps −1 , where the first uncertainty is statistical and the second is systematic. This measurement improves on the current Δ m s precision by a factor of two. We combine this result with previous LHCb measurements to determine Δ m s  = 17.7656 ± 0.0057 ps −1 , which is the legacy measurement of the original LHCb detector. The LHCb collaboration reports an improved measurement of the oscillation frequency of mesons consisting of a bottom quark and strange quark, which is then combined with previous results.

Kinematic correlations for pairs of beauty hadrons, produced in high energy proton-proton collisions, are studied. The data sample used was collected with the LHCb experiment at centre-of-mass energies of 7 and 8 TeV and corresponds to an integrated luminosity of 3 fb−1. The measurement is performed using inclusive b → J/ψX decays in the rapidity range 2 < yJ/ψ < 4.5. The observed correlations are in good agreement with theoretical predictions.

A search for \\[C\\!P\\] violation in charmless four-body decays of \\[{\\varLambda } ^0_{b} \\] and \\[{\\varXi } ^0_{b} \\] baryons with a proton and three charged mesons in the final state is performed. To cancel out production and detection charge-asymmetry effects, the search is carried out by measuring the difference between the \\[C\\!P\\] asymmetries in a charmless decay and in a decay with an intermediate charmed baryon with the same particles in the final state. The data sample used was recorded in 2011 and 2012 with the LHCb detector and corresponds to an integrated luminosity of \\[3 \\text { fb} ^{-1} \\]. A total of 18 \\[C\\!P\\] asymmetries are considered, either accounting for the full phase space of the decays or exploring specific regions of the decay kinematics. No significant \\[C\\!P\\]-violation effect is observed in any of the measurements.

A search for the rare leptonic decay \\[{{{B} ^+}} \\!\\rightarrow {\\mu ^+} {\\mu ^-} {\\mu ^+} {{\\upnu } _\\mu } \\] is performed using proton-proton collision data corresponding to an integrated luminosity of \\[4.7 \\, {\\hbox {fb}}^{-1}\\] collected by the LHCb experiment. The search is carried out in the region where the lowest of the two \\[{\\mu ^+\\mu ^-} \\] mass combinations is below \\[980 {\\mathrm {\\,MeV\\!/}c^2} \\]. The data are consistent with the background-only hypothesis and an upper limit of \\[1.6 \\times 10^{-8}\\] at 95% confidence level is set on the branching fraction in the stated kinematic region.

The branching fraction and direct \\[C\\!P\\] asymmetry of the decay \\[{{{B} ^+}} \\!\\rightarrow {{J /\\psi }} {{\\rho } ^+} \\] are measured using proton-proton collision data collected with the LHCb detector at centre-of-mass energies of 7 and 8 TeV, corresponding to a total integrated luminosity of 3\\[\\,\\text{ fb }^{-1}\\]. The following results are obtained: \\[\\begin{aligned} \\mathcal {B}({{B} ^+} \\!\\rightarrow {{J /\\psi }} {{\\rho } ^+} )&= (3.81^{+0.25}_{-0.24} \\pm 0.35) \\times 10^{-5},\\\ \\mathcal {A}^{{C\\!P}} ({{B} ^+} \\!\\rightarrow {{J /\\psi }} {{\\rho } ^+} )&= -0.045^{+0.056}_{-0.057} \\pm 0.008, \\end{aligned}\\]where the first uncertainties are statistical and the second systematic. Both measurements are the most precise to date.

The inclusive Ds± production asymmetry is measured in pp collisions collected by the LHCb experiment at centre-of-mass energies of s=7 and 8 TeV. Promptly produced Ds± mesons are used, which decay as Ds± → ϕπ±, with ϕ → K+K−. The measurement is performed in bins of transverse momentum, pT, and rapidity, y, covering the range 2.5 < pT< 25.0 GeV/c and 2.0 < y < 4.5. No kinematic dependence is observed. Evidence of nonzero Ds± production asymmetry is found with a significance of 3.3 standard deviations.

A Dalitz plot analysis of \\[{{B} ^0} \\!\\rightarrow \\eta _c(1S) {{K} ^+} {{\\pi } ^-} \\] decays is performed using data samples of pp collisions collected with the \\[\\text{ LHCb } \\] detector at centre-of-mass energies of \\[{\\sqrt{s}} =7,~8\\] and \\[13{\\,\\mathrm {Te}\\mathrm {V}} \\], corresponding to a total integrated luminosity of \\[4.7 \\,\\text{ fb }^{-1} \\]. A satisfactory description of the data is obtained when including a contribution representing an exotic \\[\\eta _c(1S) \\pi ^-\\] resonant state. The significance of this exotic resonance is more than three standard deviations, while its mass and width are \\[4096 \\pm 20~^{+18}_{-22} \\,\\mathrm {Me}\\mathrm {V} \\] and \\[152 \\pm 58~^{+60}_{-35} \\,\\mathrm {Me}\\mathrm {V} \\], respectively. The spin-parity assignments \\[J^P=0^+\\] and \\[J^{P}=1^-\\] are both consistent with the data. In addition, the first measurement of the \\[{{B} ^0} \\!\\rightarrow \\eta _c(1S) {{K} ^+} {{\\pi } ^-} \\] branching fraction is performed and gives \\[\\begin{aligned} \\displaystyle \\mathcal {B}({{B} ^0} \\!\\rightarrow \\eta _c(1S) {{K} ^+} {{\\pi } ^-} ) = (5.73 \\pm 0.24 \\pm 0.13 \\pm 0.66) \\times 10^{-4}, \\end{aligned}\\]where the first uncertainty is statistical, the second systematic, and the third is due to limited knowledge of external branching fractions.

Amplitude models are constructed to describe the resonance structure of D0→K-π+π+π- and D0→K+π-π-π+ decays using pp collision data collected at centre-of-mass energies of 7 and 8 TeV with the LHCb experiment, corresponding to an integrated luminosity of 3.0 fb-1. The largest contributions to both decay amplitudes are found to come from axial resonances, with decay modes D0→a1(1260)+K- and D0→K1(1270/1400)+π- being prominent in D0→K-π+π+π- and D0→K+π-π-π+, respectively. Precise measurements of the lineshape parameters and couplings of the a1(1260)+, K1(1270)- and K(1460)- resonances are made, and a quasi model-independent study of the K(1460)- resonance is performed. The coherence factor of the decays is calculated from the amplitude models to be RK3π=0.459±0.010(stat)±0.012(syst)±0.020(model), which is consistent with direct measurements. These models will be useful in future measurements of the unitary-triangle angle γ and studies of charm mixing and CP violation.

The decays [Formula: see text] and [Formula: see text] are observed for the first time using a data sample corresponding to an integrated luminosity of 3.0 fb[Formula: see text], collected by the LHCb experiment in proton-proton collisions at the centre-of-mass energies of 7 and 8[Formula: see text]. The branching fractions relative to that of [Formula: see text] are measured to be [Formula: see text]where the first uncertainties are statistical and the second are systematic.The decays [Formula: see text] and [Formula: see text] are observed for the first time using a data sample corresponding to an integrated luminosity of 3.0 fb[Formula: see text], collected by the LHCb experiment in proton-proton collisions at the centre-of-mass energies of 7 and 8[Formula: see text]. The branching fractions relative to that of [Formula: see text] are measured to be [Formula: see text]where the first uncertainties are statistical and the second are systematic.