Equivalence after extension for compact operators on Banach spaces

In recent years the coincidence of the operator relations equivalence after extension and Schur coupling was settled for the Hilbert space case, by showing that equivalence after extension implies equivalence after one-sided extension. In this paper we investigate consequences of equivalence after extension for compact Banach space operators. We show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension. In analogy with the necessary and sufficient conditions on the singular values for compact Hilbert space operators that are equivalent after extension, we prove the necessity of similar relationships between the $s$-numbers of two compact Banach space operators that are equivalent after extension, for arbitrary $s$-functions. We investigate equivalence after extension for operators on $\ell^{p}$-spaces. We show that two operators that act on different $\ell^{p}$-spaces cannot be equivalent after one-sided extension. Such operators can still be equivalent after extension, for instance all invertible operators are equivalent after extension, however, if one of the two operators is compact, then they cannot be equivalent after extension. This contrasts the Hilbert space case where equivalence after one-sided extension and equivalence after extension are, in fact, identical relations. Finally, for general Banach spaces $X$ and $Y$, we investigate consequences of an operator on $X$ being equivalent after extension to a compact operator on $Y$. We show that, in this case, a closed finite codimensional subspace of $Y$ must embed into $X$, and that certain general Banach space properties must transfer from $X$ to $Y$. We also show that no operator on $X$ can be equivalent after extension to an operator on $Y$, if $X$ and $Y$ are essentially incomparable Banach spaces

Global properties of eigenvalues of parametric rank one perturbations for unstructured and structured matrices II

We present in this note a correction to Theorem 17 in Ran and Wojtylak (Compl. Anal. Oper. Theory 15:44, 2021) and sharpen the estimates for eigenvalues of parametric rank one perturbations given in that theorem

Eigenvalue perturbation theory of structured real matrices and their sign characteristics under generic structured rank-one perturbations

An eigenvalue perturbation theory under rank-one perturbations is developed for classes of real matrices that are symmetric with respect to a non-degenerate bilinear form, or Hamiltonian with respect to a non-degenerate skew-symmetric form. In contrast to the case of complex matrices, the sign characteristic is a crucial feature of matrices in these classes. The behavior of the sign characteristic under generic rank-one perturbations is analyzed in each of these two classes of matrices. Partial results are presented, but some questions remain open. Applications include boundedness and robust boundedness for solutions of structured systems of linear differential equations with respect to general perturbations as well as with respect to structured rank perturbations of the coefficients

Functional interaction between BLM helicase and 53BP1 in a Chk1-mediated pathway during S-phase arrest

Bloom's syndrome is a rare autosomal recessive genetic disorder characterized by chromosomal aberrations, genetic instability, and cancer predisposition, all of which may be the result of abnormal signal transduction during DNA damage recognition. Here, we show that BLM is an intermediate responder to stalled DNA replication forks. BLM colocalized and physically interacted with the DNA damage response proteins 53BP1 and H2AX. Although BLM facilitated physical interaction between p53 and 53BP1, 53BP1 was required for efficient accumulation of both BLM and p53 at the sites of stalled replication. The accumulation of BLM/53BP1 foci and the physical interaction between them was independent of γ-H2AX. The active Chk1 kinase was essential for both the accurate focal colocalization of 53BP1 with BLM and the consequent stabilization of BLM. Once the ATR/Chk1- and 53BP1-mediated signal from replicational stress is received, BLM functions in multiple downstream repair processes, thereby fulfilling its role as a caretaker tumor suppressor

}T$.

The spectral properties of two products AB and BA of possibly unbounded operators A and B in a Banach space are considered. The results are applied in the comparison of local spectral properties of the operators

Cross-National Differences in Victimization : Disentangling the Impact of Composition and Context

Varying rates of criminal victimization across countries are assumed to be the outcome of countrylevel structural constraints that determine the supply ofmotivated o¡enders, as well as the differential composition within countries of suitable targets and capable guardianship. However, previous empirical tests of these ‘compositional’ and ‘contextual’ explanations of cross-national di¡erences have been performed upon macro-level crime data due to the unavailability of comparable individual-level data across countries. This limitation has had two important consequences for cross-national crime research. First, micro-/meso-level mechanisms underlying cross-national differences cannot be truly inferred from macro-level data. Secondly, the e¡ects of contextual measures (e.g. income inequality) on crime are uncontrolled for compositional heterogeneity. In this paper, these limitations are overcome by analysing individual-level victimization data across 18 countries from the International CrimeVictims Survey. Results from multi-level analyses on theft and violent victimization indicate that the national level of income inequality is positively related to risk, independent of compositional (i.e. micro- and meso-level) di¡erences. Furthermore, crossnational variation in victimization rates is not only shaped by di¡erences in national context, but also by varying composition. More speci¢cally, countries had higher crime rates the more they consisted of urban residents and regions with lowaverage social cohesion.

Magnetic Fields toward Ophiuchus-B Derived from SCUBA-2 Polarization Measurements

We present the results of dust emission polarization measurements of Ophiuchus-B (Oph-B) carried out using the Submillimetre Common-User Bolometer Array 2 (SCUBA-2) camera with its associated polarimeter (POL-2) on the James Clerk Maxwell Telescope in Hawaii. This work is part of the B-fields in Star-forming Region Observations survey initiated to understand the role of magnetic fields in star formation for nearby star-forming molecular clouds. We present a first look at the geometry and strength of magnetic fields in Oph-B. The field geometry is traced over ~0.2 pc, with clear detection of both of the sub-clumps of Oph-B. The field pattern appears significantly disordered in sub-clump Oph-B1. The field geometry in Oph-B2 is more ordered, with a tendency to be along the major axis of the clump, parallel to the filamentary structure within which it lies. The degree of polarization decreases systematically toward the dense core material in the two sub-clumps. The field lines in the lower density material along the periphery are smoothly joined to the large-scale magnetic fields probed by NIR polarization observations. We estimated a magnetic field strength of 630 ± 410 μG in the Oph-B2 sub-clump using a Davis–Chandrasekhar–Fermi analysis. With this magnetic field strength, we find a mass-to-flux ratio λ = 1.6 ± 1.1, which suggests that the Oph-B2 clump is slightly magnetically supercritical