Stability for manifolds of equilibrium states of generalized Birkhoff system

<正>Stability for the manifolds of equilibrium states of a generalized Birkhoff system is studied.A theorem for the stability of the manifolds of equilibrium states of the general autonomous system is used to the generalized Birkhoffian system and two propositions on the stability of the manifolds of equilibrium states of the system are obtained.An example is given to illustrate the application of the results.     

Sobolev H1 geometry of the symplectomorphism group

For a closed symplectic manifold $(M,\omega )$ with compatible Riemannian metric g we study the Sobolev $H^1$ geometry of the group of all $H^s$ diffeomorphisms on M which preserve the symplectic structure. We show that, for sufficiently large s, the $H^1$ metric admits globally defined geodesics and the corresponding exponential map is a non-linear Fredholm map of index zero. Finally, we show that the $H^1$ metric carries conjugate points via some simple examples.     

The contact Yamabe flow on K-contact manifolds

We use the contact Yamabe flow to find solutions of the contact Yamabe problem on K-contact manifolds.     

Bi-f-harmonic map equations on singly warped product manifolds

Bi-f-harmonic maps are the critical points of bi-f-energy functional. This class of maps tends to integrate bi-harmonic maps and f-harmonic maps. In this paper, we show that bi-f-harmonic maps are not only an extension of f-harmonic maps but also an extension of bi-harmonic maps, and that there should exist many examples of proper bi-f-harmonic maps.In order to find some concrete examples of proper bi-f-harmonic maps, we study the basic properties of bi-f-harmonic maps from two directions which are conformal maps between the same dimensional manifolds and some special maps from or into a warped product manifold.     

Numerical study of dielectrophoresis-modified inertial migration for overlapping sized cell separation

Circulating tumor cells (CTCs) have been proven to have significant prognostic, diagnostic, and clinical values in early-stage cancer detection and treatment. The efficient separation of CTCs from peripheral blood can ensure intact and viable CTCs and can, thus, give proper genetic characterization and drug innovation. In this study, continuous and high-throughput separation of MDA-231 CTCs from overlapping sized white blood cells (WBCs) is achieved by modifying inertial cell focusing with dielectrophoresis (DEP) in a single-stage microfluidic platform by numeric simulation. The DEP is enabled by embedding interdigitated electrodes with alternating field control on a serpentine microchannel to avoid creating two-stage separation. Rather than using the electrokinetic migration of cells which slows down the throughput, the system leverages the inertial microfluidic flow to achieve high-speed continuous separation. The cell migration and cell positioning characteristics are quantified through coupled physics analyses to evaluate the effects of the applied voltages and Reynolds numbers (Re) on the separation performance. The results indicate that the introduction of DEP successfully migrates WBCs away from CTCs and that separation of MDA-231 CTCs from similar sized WBCs at a high Re of 100 can be achieved with a low voltage of magnitude 4 ×10⁶ V/m. Additionally, the viability of MDA-231 CTCs is expected to be sustained after separation due to the short-term DEP exposure. The developed technique could be exploited to design active microchips for high-throughput separation of mixed cell beads despite their significant size overlap, using DEP-modified inertial focusing controlled simply by adjusting the applied external field.     

Inertial focusing of microparticles,bacteria, and blood in serpentine glass channels

Early detection of pathogenic microorganisms is pivotal to diagnosis and prevention of health and safety crises. Standard methods for pathogen detection often rely on lengthy culturing procedures, confirmed by biochemical assays, leading to >24 h for a diagnosis. The main challenge for pathogen detection is their low concentration within complex matrices. Detection of blood-borne pathogens via techniques such as PCR requires an initial positive blood culture and removal of inhibitory blood components, reducing its potential as a diagnostic tool. Among different label-free microfluidic techniques, inertial focusing on microscale channels holds great promise for automation, parallelization, and passive continuous separation of particles and cells. This work presents inertial microfluidic manipulation of small particles and cells (1–10 μm) in curved serpentine glass channels etched at different depths (deep and shallow designs) that can be exploited for (1) bacteria preconcentration from biological samples and (2) bacteria-blood cell separation. In our shallow device, the ability to focus Escherichia coli into the channel side streams with high recovery (89% at 2.2× preconcentration factor) could be applied for bacteria preconcentration in urine for diagnosis of urinary tract infections. Relying on differential equilibrium positions of red blood cells and E. coli inside the deep device, 97% red blood cells were depleted from 1:50 diluted blood with 54% E. coli recovered at a throughput of 0.7 mL/min. Parallelization of such devices could process relevant volumes of 7 mL whole blood in 10 min, allowing faster sample preparation for downstream molecular diagnostics of bacteria present in bloodstream.     

Anti-holomorphic involutive isometry of hyper-Kähler manifolds and branes

We study complex Lagrangian submanifolds of a compact hyper-Kähler manifold and prove two results: (a) that an involution of a hyper-Kähler manifold which is antiholomorphic with respect to one complex structure and which acts non-trivially on the corresponding symplectic form always has a fixed point locus which is complex Lagrangian with respect to one of the other complex structures, and (b) there exist Lagrangian submanifolds which are complex with respect to one complex structure and are not the fixed point locus of any involution which is anti-holomorphic with respect to one of the other complex structures.     

Strichartz Estimates for Non-Elliptic Schrödinger Equations on Compact Manifolds

In this note we consider the Schrödinger equation on compact manifolds equipped with possibly degenerate metrics. We prove Strichartz estimates with a loss of derivatives. The rate of loss of derivatives depends on the degeneracy of metrics. For the non-degenerate case we obtain, as an application of the main result, the same Strichartz estimates as that in the elliptic case. This extends Strichartz estimates for Riemannian metrics proved by Burq-Gérard-Tzvetkov to the non-elliptic case and improves the result by Salort for the degenerate case. We also investigate the optimality of the result for the case on 𝕊³ × 𝕊³.