Young's integral inequality with upper and lower bounds

Young's integral inequality is reformulated with upper and lower bounds for the remainder. The new inequalities improve Young's integral inequality on all time scales, such that the case where equality holds becomes particularly transparent in this new presentation. The corresponding results for difference equations are given, and several examples are included. We extend these results to piecewise-monotone functions as well.Comment: 8 page

Raman scattering in correlated thin films as a probe of chargeless surface states

Several powerful techniques exist to detect topologically protected surface states of weakly-interacting electronic systems. In contrast, surface modes of strongly interacting systems which do not carry electric charge are much harder to detect. We propose resonant light scattering as a means of probing the chargeless surface modes of interacting quantum spin systems, and illustrate its efficacy by a concrete calculation for the 3D hyperhoneycomb Kitaev quantum spin liquid phase. We show that resonant scattering is required to efficiently couple to this model's sublattice polarized surface modes, comprised of emergent Majorana fermions that result from spin fractionalization. We demonstrate that the low-energy response is dominated by the surface contribution for thin films, allowing identification and characterization of emergent topological band structures.Comment: 7 pages, 4 figures; added supplemental materia

Theory of Raman response in three-dimensional Kitaev spin liquids: application to β−\beta- and γ−\gamma-Li2_2IrO3_3 compounds

We calculate the Raman response for the Kitaev spin model on the $\mathcal{H}$-$0$, $\mathcal{H}$-$1$, and $\mathcal{H}$-$\infty$ harmonic honeycomb lattices. We identify several quantitative features in the Raman spectrum that are characteristic of the spin liquid phase. Unlike the dynamical structure factor, which probes both the Majorana spinons and flux excitations that emerge from spin fractionalization, the Raman spectrum in the Kitaev models directly probes a density of states of pairs of fractional, dispersing Majorana spinons. As a consequence, the Raman spectrum in all these models is gapless for sufficiently isotropic couplings, with a low-energy power law that results from the Fermi lines (or points) of the dispersing Majorana spinons. We show that the polarization dependence of the Raman spectrum contains crucial information about the symmetry of the ground state. We also discuss to what extent the features of the Raman response that we find reflect generic properties of the spin liquid phase, and comment on their possible relevance to $\alpha-$, $\beta-$ and $\gamma-$Li$_2$IrO$_3$ compounds.Comment: 19 pages, 10 figures. VERSION 2: Corrected Figure 5 and fixed inconsistencies between A and B chain-labelings. Also- a few typos and two new ref

Resonant Raman scattering theory for Kitaev models and their Majorana fermion boundary modes

We study the inelastic light scattering response in two- (2D) and three-dimensional (3D) Kitaev spin-liquid models with \ms band structures in the symmetry classes BDI and D leading to protected gapless surface modes. We present a detailed calculation of the resonant Raman/Brillouin scattering vertex relevant to iridate and ruthenate compounds whose low-energy physics is believed to be proximate to these spin-liquid phases. In the symmetry class BDI, we find that while the resonant scattering on thin films can detect the gapless boundary modes of spin liquids, the non-resonant processes do not couple to them. For the symmetry class D, however, we find that the coupling between both types of light-scattering processes and the low-energy surface states is strongly suppressed. Additionally, we describe the effect of weak time-reversal symmetry breaking perturbations on the bulk Raman response of these systems.Comment: 23 pages, 20 figures, 4 appendices, 2 ancillary file

Clinical outcomes and patient-matched molecular composition of relapsed medulloblastoma

© 2021 by American Society of Clinical Oncology. Creative Commons Attribution Non-Commercial No Derivatives 4.0 License: https://creativecommons.org/licenses/by-nc-nd/4.0/Purpose: We sought to investigate clinical outcomes of relapsed medulloblastoma and to compare molecular features between patient-matched diagnostic and relapsed tumors. Methods: Children and infants enrolled on either SJMB03 (NCT00085202) or SJYC07 (NCT00602667) trials who experienced medulloblastoma relapse were analyzed for clinical outcomes, including anatomic and temporal patterns of relapse and postrelapse survival. A largely independent, paired molecular cohort was analyzed by DNA methylation array and next-generation sequencing. Results: A total of 72 of 329 (22%) SJMB03 and 52 of 79 (66%) SJYC07 patients experienced relapse with significant representation of Group 3 and wingless tumors. Although most patients exhibited some distal disease (79%), 38% of patients with sonic hedgehog tumors experienced isolated local relapse. Time to relapse and postrelapse survival varied by molecular subgroup with longer latencies for patients with Group 4 tumors. Postrelapse radiation therapy among previously nonirradiated SJYC07 patients was associated with long-term survival. Reirradiation was only temporizing for SJMB03 patients. Among 127 patients with patient-matched tumor pairs, 9 (7%) experienced subsequent nonmedulloblastoma CNS malignancies. Subgroup (96%) and subtype (80%) stabilities were largely maintained among the remainder. Rare subgroup divergence was observed from Group 4 to Group 3 tumors, which is coincident with genetic alterations involving MYC, MYCN, and FBXW7. Subgroup-specific patterns of alteration were identified for driver genes and chromosome arms. Conclusion: Clinical behavior of relapsed medulloblastoma must be contextualized in terms of up-front therapies and molecular classifications. Group 4 tumors exhibit slower biological progression. Utility of radiation at relapse is dependent on patient age and prior treatments. Degree and patterns of molecular conservation at relapse vary by subgroup. Relapse tissue enables verification of molecular targets and identification of occult secondary malignancies.info:eu-repo/semantics/publishedVersio