A threshold phenomenon for embeddings of H0mH^m_0 into Orlicz spaces

We consider a sequence of positive smooth critical points of the Adams-Moser-Trudinger embedding of $H^m_0$ into Orlicz spaces. We study its concentration-compactness behavior and show that if the sequence is not precompact, then the liminf of the $H^m_0$-norms of the functions is greater than or equal to a positive geometric constant.Comment: 14 Page

Quantization for an elliptic equation of order 2m with critical exponential non-linearity

On a smoothly bounded domain $\Omega\subset\R{2m}$ we consider a sequence of positive solutions $u_k\stackrel{w}{\rightharpoondown} 0$ in $H^m(\Omega)$ to the equation $(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}$ subject to Dirichlet boundary conditions, where $0<\lambda_k\to 0$. Assuming that $$\Lambda:=\lim_{k\to\infty}\int_\Omega u_k(-\Delta)^m u_k dx<\infty,$$ we prove that $\Lambda$ is an integer multiple of \Lambda_1:=(2m-1)!\vol(S^{2m}), the total $Q$-curvature of the standard $2m$-dimensional sphere.Comment: 33 page

Renormalization and blow up for charge one equivariant critical wave maps

We prove the existence of equivariant finite time blow up solutions for the wave map problem from 2+1 dimensions into the 2-sphere. These solutions are the sum of a dynamically rescaled ground-state harmonic map plus a radiation term. The local energy of the latter tends to zero as time approaches blow up time. This is accomplished by first "renormalizing" the rescaled ground state harmonic map profile by solving an elliptic equation, followed by a perturbative analysis

Collision cross sections of high-mannose N-glycans in commonly observed adduct states – identification of gas-phase conformers unique to [M − H]^- ions

We report collision cross sections (CCS) of high-mannose N-glycans as [M + Na]+, [M + K]+, [M + H]+, [M + Cl]-, [M + H2PO4]- and [M − H]- ions, measured by drift tube (DT) ion mobility-mass spectrometry (IM-MS) in helium and nitrogen gases. Further analysis using traveling wave (TW) IM-MS reveal the existence of distinct conformers exclusive to [M − H]- ions

Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system

We describe the asymptotic behavior as time goes to infinity of solutions of the 2 dimensional corotational wave map system and of solutions to the 4 dimensional, radially symmetric Yang-Mills equation, in the critical energy space, with data of energy smaller than or equal to a harmonic map of minimal energy. An alternative holds: either the data is the harmonic map and the soltuion is constant in time, or the solution scatters in infinite time

N-glycan microheterogeneity regulates interactions of plasma proteins

Altered glycosylation patterns of plasma proteins are associated with autoimmune disorders and pathogenesis of various cancers. Elucidating glycoprotein microheterogeneity and relating subtle changes in the glycan structural repertoire to changes in protein–protein, or protein–small molecule interactions, remains a significant challenge in glycobiology. Here, we apply mass spectrometry-based approaches to elucidate the global and site-specific microheterogeneity of two plasma proteins: α1-acid glycoprotein (AGP) and haptoglobin (Hp). We then determine the dissociation constants of the anticoagulant warfarin to different AGP glycoforms and reveal how subtle N-glycan differences, namely, increased antennae branching and terminal fucosylation, reduce drug-binding affinity. Conversely, similar analysis of the haptoglobin–hemoglobin (Hp–Hb) complex reveals the contrary effects of fucosylation and N-glycan branching on Hp–Hb interactions. Taken together, our results not only elucidate how glycoprotein microheterogeneity regulates protein–drug/protein interactions but also inform the pharmacokinetics of plasma proteins, many of which are drug targets, and whose glycosylation status changes in various disease states

KAM for the quantum harmonic oscillator

In this paper we prove an abstract KAM theorem for infinite dimensional Hamiltonians systems. This result extends previous works of S.B. Kuksin and J. P\"oschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an application we show that some 1D nonlinear Schr\"odinger equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schr\"odinger equations with the harmonic potential and a quasi periodic in time potential.Comment: 54 pages. To appear in Comm. Math. Phy

On a functional satisfying a weak Palais-Smale condition

In this paper we study a quasilinear elliptic problem whose functional satisfies a weak version of the well known Palais-Smale condition. An existence result is proved under general assumptions on the nonlinearities.Comment: 18 page

On Singularity formation for the L^2-critical Boson star equation

We prove a general, non-perturbative result about finite-time blowup solutions for the $L^2$-critical boson star equation $i\partial_t u = \sqrt{-\Delta+m^2} \, u - (|x|^{-1} \ast |u|^2) u$ in 3 space dimensions. Under the sole assumption that the solution blows up in $H^{1/2}$ at finite time, we show that $u(t)$ has a unique weak limit in $L^2$ and that $|u(t)|^2$ has a unique weak limit in the sense of measures. Moreover, we prove that the limiting measure exhibits minimal mass concentration. A central ingredient used in the proof is a "finite speed of propagation" property, which puts a strong rigidity on the blowup behavior of $u$. As the second main result, we prove that any radial finite-time blowup solution $u$ converges strongly in $L^2$ away from the origin. For radial solutions, this result establishes a large data blowup conjecture for the $L^2$-critical boson star equation, similar to a conjecture which was originally formulated by F. Merle and P. Raphael for the $L^2$-critical nonlinear Schr\"odinger equation in [CMP 253 (2005), 675-704]. We also discuss some extensions of our results to other $L^2$-critical theories of gravitational collapse, in particular to critical Hartree-type equations.Comment: 24 pages. Accepted in Nonlinearit