Convergence of finite volume scheme for degenerate parabolic problem with zero flux boundary condition

This note is devoted to the study of the finite volume methods used in the discretization of degenerate parabolic-hyperbolic equation with zero-flux boundary condition. The notion of an entropy-process solution, successfully used for the Dirichlet problem, is insufficient to obtain a uniqueness and convergence result because of a lack of regularity of solutions on the boundary. We infer the uniqueness of an entropy-process solution using the tool of the nonlinear semigroup theory by passing to the new abstract notion of integral-process solution. Then, we prove that numerical solution converges to the unique entropy solution as the mesh size tends to 0

A theory of L1L^1-dissipative solvers for scalar conservation laws with discontinuous flux

We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are certain piecewise constant stationary weak solutions. We refer to such a family as a "germ". It is well known that (CL) admits many different $L^1$ contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the anishing viscosity" germ, which is a way to express the "$\Gamma$-condition" of Diehl. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line $x=0$ (in the spirit of Vol'pert) and in the form of a family of global entropy inequalities (following Kruzhkov and Carrillo). We characterize those germs that lead to the $L^1$-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes

Computing the first eigenpair of the p-Laplacian via inverse iteration of sublinear supersolutions

We introduce an iterative method for computing the first eigenpair $(\lambda_{p},e_{p})$ for the $p$-Laplacian operator with homogeneous Dirichlet data as the limit of $(\mu_{q,}u_{q})$ as $q\rightarrow p^{-}$, where $u_{q}$ is the positive solution of the sublinear Lane-Emden equation $-\Delta_{p}u_{q}=\mu_{q}u_{q}^{q-1}$ with same boundary data. The method is shown to work for any smooth, bounded domain. Solutions to the Lane-Emden problem are obtained through inverse iteration of a super-solution which is derived from the solution to the torsional creep problem. Convergence of $u_{q}$ to $e_{p}$ is in the $C^{1}$-norm and the rate of convergence of $\mu_{q}$ to $\lambda_{p}$ is at least $O(p-q)$. Numerical evidence is presented.Comment: Section 5 was rewritten. Jed Brown was added as autho

Observation of the Decay Λ0b→Λ+cτ−¯ν

The first observation of the semileptonic b-baryon decay Λb0→Λc+τ-ν¯τ, with a significance of 6.1σ, is reported using a data sample corresponding to 3 fb-1 of integrated luminosity, collected by the LHCb experiment at center-of-mass energies of 7 and 8 TeV at the LHC. The τ- lepton is reconstructed in the hadronic decay to three charged pions. The ratio K=B(Λb0→Λc+τ-ν¯τ)/B(Λb0→Λc+π-π+π-) is measured to be 2.46±0.27±0.40, where the first uncertainty is statistical and the second systematic. The branching fraction B(Λb0→Λc+τ-ν¯τ)=(1.50±0.16±0.25±0.23)% is obtained, where the third uncertainty is from the external branching fraction of the normalization channel Λb0→Λc+π-π+π-. The ratio of semileptonic branching fractions R(Λc+)B(Λb0→Λc+τ-ν¯τ)/B(Λb0→Λc+μ-ν¯μ) is derived to be 0.242±0.026±0.040±0.059, where the external branching fraction uncertainty from the channel Λb0→Λc+μ-ν¯μ contributes to the last term. This result is in agreement with the standard model prediction

A study of CP violation in the decays B±→[K+K-π+π-]Dh± (h= K, π) and B±→[π+π-π+π-]Dh±

The first study of CP violation in the decay mode B±→[K+K-π+π-]Dh± , with h= K, π , is presented, exploiting a data sample of proton–proton collisions collected by the LHCb experiment that corresponds to an integrated luminosity of 9 \,fb - 1 . The analysis is performed in bins of phase space, which are optimised for sensitivity to local CP asymmetries. CP -violating observables that are sensitive to the angle γ of the Unitarity Triangle are determined. The analysis requires external information on charm-decay parameters, which are currently taken from an amplitude analysis of LHCb data, but can be updated in the future when direct measurements become available. Measurements are also performed of phase-space integrated observables for B±→[K+K-π+π-]Dh± and B±→[π+π-π+π-]Dh± decays

Measurement of the photon polarization in Λb→Λγ\Lambda_b \to \Lambda \gamma decays

The photon polarization in $b \to s \gamma$ transitions is measured for the first time in radiative b-baryon decays exploiting the unique spin structure of $\Lambda_b \to \Lambda \gamma$ decays. A data sample corresponding to an integrated luminosity of $6\;fb^{-1}$ collected by the LHCb experiment in $pp$ collisions at a center-of-mass energy of $13\;TeV$ is used. The photon polarization is measured to be $\alpha_{\gamma}= 0.82^{\,+\,0.17\,+\,0.04}_{\,-\,0.26\,-\,0.13}$, where the first uncertainty is statistical and the second systematic. This result is in agreement with the Standard Model prediction and previous measurements in b-meson decays. Charge-parity breaking effects are studied for the first time in this observable and found to be consistent with $CP$ symmetry.Comment: All figures and tables, along with machine-readable versions and any supplementary material and additional information, are available at https://cern.ch/lhcbproject/Publications/p/LHCb-PAPER-2021-030.html (LHCb public pages