CR-Invariants and the Scattering Operator for Complex Manifolds with Boundary

The purpose of this paper is to describe certain CR-covariant differential operators on a strictly pseudoconvex CR manifold $M$ as residues of the scattering operator for the Laplacian on an ambient complex K\"{a}hler manifold $X$ having $M$ as a `CR-infinity.' We also characterize the CR $Q$-curvature in terms of the scattering operator. Our results parallel earlier results of Graham and Zworski \cite{GZ:2003}, who showed that if $X$ is an asymptotically hyperbolic manifold carrying a Poincar\'{e}-Einstein metric, the $Q$-curvature and certain conformally covariant differential operators on the `conformal infinity' $M$ of $X$ can be recovered from the scattering operator on $X$. The results in this paper were announced in \cite{HPT:2006}.Comment: 32 page

Dependence of the density of states outer measure on the potential for deterministic Schr\"odinger operators on graphs with applications to ergodic and random models

We continue our study of the dependence of the density of states measure and related spectral functions of Schr\"odinger operators on the potential. Whereas our earlier work focused on random Schr\"odinger operators, we extend these results to Schr\"odinger operators on infinite graphs with deterministic potentials and ergodic potentials, and improve our results for random potentials. In particular, we prove the Lipschitz continuity of the DOSm for random Schr\"odinger operators on the lattice, recovering results of \cite{kachkovskiy, shamis}. For our treatment of deterministic potentials, we first study the density of states outer measure (DOSoM), defined for all Schr\"odinger operators, and prove a deterministic result of the modulus of continuity of the DOSoM with respect to the potential. We apply these results to Schr\"odinger operators on the lattice $Z^d$ and the Bethe lattice. In the former case, we prove the Lipschitz continuity of the DOSoM, and in the latter case, we prove that the DOSoM is $\frac{1}{2}$-log-H\"older continuous. Our technique combines the abstract Lipschitz property of one-parameter families of self-adjoint operators with a new finite-range reduction that allows us to study the dependency of the DOSoM and related functions on only finitely-many variables and captures the geometry of the graph at infinity.Comment: Related to arXiv:1804.02444 and arXiv:1904.01118 by the authors; New appendices C and D are added and typos corrected. Appendix C discusses inequalities between the metric of weak convergence of measures and the Kantorovich-Rubinstein-Wasserstein metric. Appendix D presents results on continuity of the Hausdorff distance between two spectra with respect to the potential

Using Game of Thrones to teach neuroscience and neuropharmacology during lockdown

Peer reviewedPublisher PD

Separation of variables in perturbed cylinders

We study the Laplace operator subject to Dirichlet boundary conditions in a two-dimensional domain that is one-to-one mapped onto a cylinder (rectangle or infinite strip). As a result of this transformation the original eigenvalue problem is reduced to an equivalent problem for an operator with variable coefficients. Taking advantage of the simple geometry we separate variables by means of the Fourier decomposition method. The ODE system obtained in this way is then solved numerically yielding the eigenvalues of the operator. The same approach allows us to find complex resonances arising in some non-compact domains. We discuss numerical examples related to quantum waveguide problems.Comment: LaTeX 2e, 18 pages, 6 figure

CD8+ immunodominance among Epstein-Barr virus lytic cycle antigens directly reflects the efficiency of antigen presentation in lytically infected cells

Antigen immunodominance is an unexplained feature of CD8+ T cell responses to herpesviruses, which are agents whose lytic replication involves the sequential expression of immediate early (IE), early (E), and late (L) proteins. Here, we analyze the primary CD8 response to Epstein-Barr virus (EBV) infection for reactivity to 2 IE proteins, 11 representative E proteins, and 10 representative L proteins, across a range of HLA backgrounds. Responses were consistently skewed toward epitopes in IE and a subset of E proteins, with only occasional responses to novel epitopes in L proteins. CD8+ T cell clones to representative IE, E, and L epitopes were assayed against EBV-transformed lymphoblastoid cell lines (LCLs) containing lytically infected cells. This showed direct recognition of lytically infected cells by all three sets of effectors but at markedly different levels, in the order IE > E ≫ L, indicating that the efficiency of epitope presentation falls dramatically with progress of the lytic cycle. Thus, EBV lytic cycle antigens display a hierarchy of immunodominance that directly reflects the efficiency of their presentation in lytically infected cells; the CD8+ T cell response thereby focuses on targets whose recognition leads to maximal biologic effect

Morphological evolution and galactic sizes in the L-Galaxies SA model

In this work we update theL-Galaxiessemi-analytic model (SAM) to better follow thephysical processes responsible for the growth of bulges via disc instabilities (leading to pseudo-bulges) and mergers (leading to classical bulges). We address the former by considering thecontribution of both stellar and gaseous discs in the stability of the galaxy, and we update thelatter by including dissipation of energy in gas-rich mergers. Furthermore, we introduce angularmomentum losses during cooling and find that an accurate match to the observed correlationbetween stellar disc scale length and mass atz∼0.0requires that the gas loses 20%of its initialspecific angular momentum to the corresponding dark matter halo during the formation of thecold gas disc. We reproduce the observed trends between the stellar mass and specific angularmomentum for both disc- and bulge-dominated galaxies, with the former rotating faster thanthe latter of the same mass. We conclude that a two-component instability recipe provides amorphologically diverse galaxy sample which matches the observed fractional breakdown ofgalaxies into different morphological types. This recipe also enables us to obtain an excellent fitto the morphology-mass relation and stellar mass function of different galactic types. Finally, we find that energy dissipation during mergers reduces the merger remnant sizes and allowsus to match the observed mass-size relation for bulge-dominated system

Singular Modes of the Electromagnetic Field

We show that the mode corresponding to the point of essential spectrum of the electromagnetic scattering operator is a vector-valued distribution representing the square root of the three-dimensional Dirac's delta function. An explicit expression for this singular mode in terms of the Weyl sequence is provided and analyzed. An essential resonance thus leads to a perfect localization (confinement) of the electromagnetic field, which in practice, however, may result in complete absorption.Comment: 14 pages, no figure