Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II

We continue our investigation on the newly introduced concept of the τ function [1], associated with monodromy preserving deformations of a linear differential equation $\text{dY}\text{dx}\text{= A(x)Y}$. In this paper a series of τ functions parametrized by integers are introduced. Their ratios to the original τ function are then shown to be explicit rational expressions in terms of the coefficients of A(x). We point out, and follow where possible, a close analogy between the theory of τ functions and that of classical theta functions.     

Sets of rigged paths with Virasoro characters

Let {M _r,s ^(p,p′)}_{1≤r≤p−1,1≤s≤p′−1} be the irreducible Virasoro modules in the (p,p′)-minimal series. In our previous paper, we have constructed a monomial basis of ⊕ _r=1 ^p−1 M _r,s ^(p,p′) in the case 1<p′/p<2. By ‘monomials’ we mean vectors of the form , where φ _−n ^(r′,r):M _r,s ^(p,p′)→M _r′,s ^(p,p′) are the Fourier components of the (2,1)-primary field and |r ₀,s〉 is the highest weight vector of . In this article, we introduce for all p<p′ with p≥3 and s=1 a subset of such monomials as a conjectural basis of ⊕ _r=1 ^p−1 M _r,1^(p,p′). We prove that the character of the combinatorial set labeling these monomials coincides with the character of the corresponding Virasoro module. We also verify the conjecture in the case p=3.      

Anomalous fluctuations in sliding motion of cytoskeletal filaments driven by molecular motors: model simulations

It has been found in in vitro experiments that cytoskeletal filaments driven by molecular motors show finite diffusion in sliding motion even in the long filament limit [Imafuku, Y. et al. Biophys. J. 1996, 70, 878-886. Noda, N. et al. Biophysics 2005, 1, 45-53]. This anomalous fluctuation can be evidence for cooperativity among the motors in action because fluctuation should be averaged out for a long filament if the action of each motor is independent. In order to understand the nature of the fluctuation in molecular motors, we perform numerical simulations and analyze velocity correlation in three existing models that are known to show some kind of cooperativity and/or large diffusion coefficient, i.e., the Sekimoto-Tawada model [Sekimoto, K.; Tawada, K. Phys. Rev. Lett. 1995, 75, 180], the Prost model [Prost, J. et al. Phys. Rev. Lett. 1994, 72, 2652], and the Duke model [Duke, T. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 2770]. It is shown that the Prost model and the Duke model do not give a finite diffusion in the long filament limit, in spite of the collective action of motors. On the other hand, the Sekimoto-Tawada model has been shown to give a diffusion coefficient that is independent of filament length, but it comes from the long time correlation whose time scale is proportional to filament length, and our simulations show that such a long correlation time conflicts with the experimental time scales. We conclude that none of the three models represent experimental findings. In order to explain the observed anomalous diffusion, we have to search for a mechanism that will allow both the amplitude and the time scale of the velocity correlation to be independent of the filament length.     

Preparation and applications of a variety of fluoroalkyl end-capped oligomer/hydroxyapatite composites

A variety of fluoroalkyl end-capped oligomers were applied to the preparation of fluorinated oligomer/hydroxyapatite (HAp) composites (particle size: 38-356 nm), which exhibit a good dispersibility in water and traditional organic solvents. These fluoroalkyl end-capped oligomer/HAp composites were easily prepared by the reactions of disodium hydrogen phosphate and calcium chloride in the presence of self-assembled molecular aggregates formed by fluoroalkyl end-capped oligomers in aqueous solutions. In these fluorinated HAp composites thus obtained, fluoroalkyl end-capped acrylic acid oligomers and 2-methacryloyloxyethanesulfonic acid oligomer/HAp nanocomposites afforded transparent colorless solutions toward water; however, fluoroalkyl end-capped N,N-dimethylacrylamide oligomer and acryloylmorpholine oligomer were found to afford transparent colorless solutions with trace amounts of white-colored HAp precipitants under similar conditions. HAp could be encapsulated more effectively into fluorinated 2-methacryloyloxyethanesulfonic acid oligomeric aggregate cores to afford colloidal stable fluorinated oligomer/HAp composites, compared to that of fluorinated acrylic acid oligomers. These fluorinated oligomer/HAp composites were applied to the surface modification of glass and PVA to exhibit a good oleophobicity imparted by fluorine. HAp formation was newly observed on the modified polyethylene terephthalate film surface treated with fluorinated 2-methacryloyloxyethanesulfonic acid oligomers and acrylic acid oligomer/HAp composites by soaking these films into the simulated body fluid.     

Creation of coating surfaces possessing superhydrophobic and superoleophobic characteristics with fluoroalkyl end-capped vinyltrimethoxysilane oligomeric nanocomposites having biphenylene segments

Fluoroalkyl end-capped vinyltrimethoxysilane oligomeric nanocomposites having biphenylene units [R(F)-(VM-SiO(2))(n)-R(F)/Ar-SiO(2)] were prepared by the sol-gel reaction of the corresponding oligomer [R(F)-(VM)(n)-R(F)] with 4,4'-bis(triethoxysilyl)-1,1'-biphenyl [Ar-Si(OEt)(3)] under alkaline conditions. R(F)-(VM-SiO(2))(n)-R(F)/Ar-SiO(2) nanocomposites were applied to the surface modification of PMMA to exhibit not only a good oleophobicity imparted by fluorine but also a fluorescent emission ability on the surface. Methanol sol solutions of R(F)-(VM-SiO(2))(n)-R(F)/Ar-SiO(2) nanocomposites were effective for the surface modification of glass through the dipping technique to exhibit good oleophobicity with superhydrophobicity on the modified glass surface. On the other hand, 1,2-dichloroethane sol solutions enabled R(F)-(VM-SiO(2))(n)-R(F)/Ar-SiO(2) nanocomposites to exhibit both superhydrophobic and superoleophobic characteristics on the modified surface through dipping the glass in these sol solutions.     

Device for co-culture of sympathetic neurons and cardiomyocytes using microfabrication

Rat superior cervical ganglion (SCG) neurons and ventricular myocytes (VMs) were co-cultured separately in a minichamber placed on a microelectrode-array (MEA) substrate. The minichamber, fabricated photolithographically using polydimethylsiloxane (PDMS), had 2 compartments, 16 microcompartments and 8 microconduits. The SCG neurons were seeded into one of the compartments and all of the microcompartments using a glass pipette controlled by a micromanipulator and a microinjector. The VMs were seeded into the other compartment. Three days after seeding of the VMs, the SCG neurons were still confined to one compartment and all of the microcompartments, and the neurites of the SCG neurons had connected with the VMs via the microconduits. Constant-voltage stimulation, using a train of biphasic square pulses (1 ms at +1 V, followed by -1 ms at 1 V), was applied to the SCG neurons in the microcompartments using 16 electrodes. Evoked responses were observed in several electrodes while electrical stimulation was applied to the SCG neurons. Two-way analysis of variance (ANOVA) revealed that the frequency of the stimulation pulses had significant effects in increasing the beat rate of the VMs, and that the interaction between the frequency and the number of the pulses also had a significant effect on the ratio. No significant increases in the beat rate were observed when propranolol, a β-adrenergic receptor antagonist, was added to the culture medium. These results suggest that synaptic pathways were formed between the SCG neurons and the VMs, and that this co-culture device can be utilized for studies of network-level interactions between sympathetic neurons and cardiomyocytes.     

Two-photon-excited fluorescence from silicate glass containing tantalum ions pumped by a near-infrared femtosecond pulsed laser

We report a novel phenomenon in sodium-calcium-silicate glass doped with Ta(5+). Under irradiation with a 780 nm femtosecond pulsed laser, strong blue emission centered at about 420 nm could be observed. The spectral characteristics are similar to those pumped by ultraviolet photons. The log-log correlation between integrated emission intensity and pump power reveals that a two-photon absorption process is involved in the phenomenon. It is suggested that the presence of localized Ta(5+)5d(0) energy levels is responsible for the appearance of the blue emission. The results indicate that transition metal ions without d electrons play an important role in fields of optics when embedded into glass hosts.     

Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III

A unified treatment of monodromy and spectrum preserving deformations is presented. In particular a general procedure is described to reduce the latter into the former consistently. The concept of the τ-function, previously introduced for the former [2], is extended to the isospectral context. It is shown that the general monodromy and spectrum preserving deformation equations can be written as Hirota's bilinear differential equations by using the τ-functions as dependent variables.