Convergence of Riemannian Manifolds and Laplace Operators, II

We study spectral convergence of compact Riemannian manifolds or more generally certain Dirichlet spaces, obtaining some compactness results on harmonic functions and harmonic maps. Mathematics Subject Classifications (2000) 53C21, 58D17, 58J50. Atsushi Kasue: Partly supported by the Grant-in-Aid for Scientific Research (B) No. 15340053 of the Japan Society for the Promotion of Science.     

Lemniscates and inequalities for the logarithmic capacities of continua

It is shown that if P(z) = z ⁿ + ? is a polynomial with connected lemniscate E(P) = {z: ¦P(z)¦ ≤ 1} and m critical points, then, for any n? m+1 points on the lemniscate E(P), there exists a continuum γ ? E(P) of logarithmic capacity cap γ ≤ 2^?1/n which contains these points and all zeros and critical points of the polynomial. As corollaries, estimates for continua of minimum capacity containing given points are obtained.     

Random walks generated by affine mappings

This article is concerned with Markov chains on ^m constructed by randomly choosing an affine map at each stage, and then making the transition from the current point to its image under this map. The distribution of the random affine map can depend on the current point (i.e., state of the chain). Sufficient conditions are given under which this chain is ergodic.     

Structural determinants of steroids for cytochrome P450 3A4-mediated metabolism

Cytochrome P450 3A4 (CYP3A4), a major member of cytochrome P450 isoenzymes, metabolizes the majority of steroids in 6β-position. For the purpose of determining requisite structural features of a series of structurally related steroids for CYP3A4-mediated metabolism, three-dimensional pharmacophore modeling as well as electrotopological state map were conducted for 15 steroids. Though prior studies speculated that the chemical reactivity of the allylic 6β-position might have a greater influence on CYP3A4 selective 6-hydroxylation than steric constraints in the enzyme, our results reveal that for CYP3A4 steroidal substrates, it is not the chemical reactivity of atoms at 6β-site, but the pharmacophoric features, i.e. the two hydrophobic rings together with two H-bond donors, that act as the key factors responsible for determining the CYP3A4 selective 6-hydroxylation of steroids.     

Dual microband electrodes: current distributions and diffusion layer ‘titrations’. Implications for electroanalytical measurements

The simulation of transport to double microband electrodes in generator–collector mode is reported focusing especially on the ‘titration curve’ approach to electroanalysis in which a titrant is electrogenerated from a redox active precursor on the generator electrode and reacts homogeneously with the target analyte. The current on the detector electrode reflects the amount of titrant ‘surviving’ passage between the two electrodes. The form of the titration curve – plots of detector current as a function of generator current – is shown to be highly sensitive to the electrode kinetics of the redox couple driven at the generator electrode. Accordingly the naïve use of such methodology for analysis without accompanying simulation and kinetic analysis is fraught with danger. Use of the conformal mapping approach in combination with the ADI method for investigation of the ‘titration’ current distributions at the double band system gives fast and precise simulation of this and similar problems. Convergence analysis is described which allows for the automatic selection of the simulation grid size so as to obtain a chosen accuracy (for example 1%) of the current for all experimentally meaningful values of the geometrical and physico-chemical parameters of the system to be investigated.     

Velocity map imaging of the photolysis of n-C4H9Br in UV region

Using velocity map ion imaging technique, the photodissociation of n-C₄H₉Br in the wavelength range 231–267 nm was studied. The results and our ab initio calculations indicated that the absorption of n-C₄H₉Br in the investigated region originated from the excitations to the lowest three repulsive states, as assigned as 1A″, 2A′ and 3A′ in C_s symmetry. Dissociations occurred on the PES surfaces of the three states, terminating in C₄H₉+Br (²P_3/2) or C₄H₉ + Br^* (²P_1/2) as two channels, and being impacted by an avoided crossing between the PES surfaces of the 2A′ and 3A′ states. The transition dipole to the 1A″ state was perpendicular to the symmetry plane, so perpendicular to the C–Br bond. The transitions to the 3A′ state was polarized parallel to the symmetry plane, and also parallel to the C–Br bond. While the transition dipole to the 2A′ state was in the symmetry plane, but formed an angle of about 53.1° with the C–Br bond. We have also determined the avoided crossing probabilities, which affected the relative fractions of the individual pathways, for the photolysis of n-C₄H₉Br near 234 nm and 267 nm.     

Separation of peptides and proteins by capillary electrophoresis using acidic buffers containing tetraalkylammonium cations and cyclodextrins

A method for improving separations of peptides and other positively charged species in capillary zone electrophoresis with untreated capillaries using acidic buffers containing tetraalkylammonium cations is described. Tetramethylammonium and tetrabutylammonium cations dynamically modify the capillary surface, leading to a reversal in the direction of the electroosmotic flow. As a result, the adsorption of positively charged peptides and proteins is minimized, and resolution and peak capacity are improved as the migration of cationic analytes is counterbalanced by the electroosmotic flow. The combining effect of reversing electroosmotic flow and cyclodextrin inclusion complexation on separations of closely related peptides and a protein mixture, as well as tryptic digest of hemoglobin is demonstrated.     

On Tutte polynomial expansion formulas in perspectives of matroids and oriented matroids

We introduce the active partition of the ground set of an oriented matroid perspective (or morphism, or quotient, or strong map) on a linearly ordered ground set. The reorientations obtained by arbitrarily reorienting parts of the active partition share the same active partition. This yields an equivalence relation for the set of reorientations of an oriented matroid perspective, whose classes are enumerated by coefficients of the Tutte polynomial, and a remarkable partition of the set of reorientations into Boolean lattices, from which we get a short direct proof of a 4-variable expansion formula for the Tutte polynomial in terms of orientation activities. This formula was given in the last unpublished preprint by Michel Las Vergnas; the above equivalence relation and notion of active partition generalize a former construction in oriented matroids by Michel Las Vergnas and the author; and the possibility of such a proof technique in perspectives was announced in the aforementioned preprint. We also briefly highlight how the 5-variable expansion of the Tutte polynomial in terms of subset activities in matroid perspectives comes in a similar way from the known partition of the power set of the ground set into Boolean lattices related to subset activities (and we complete the proof with a property which was missing in the literature). In particular, the paper applies to matroids and oriented matroids on a linearly ordered ground set, and applies to graph and directed graph homomorphisms on a linearly ordered edge-set.     

Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells北大核心CSCD

The nonlinear vibration responses of functionally graded materials (FGMs) shells with different cone angles under external loads were studied. Firstly, the Voigt model was employed to describe the physical properties along the thickness direction of FGMs conical shells. Then, the motion equations were derived based on the 1st-order shear deformation theory, the von Kármán geometric nonlinearity and Hamilton’s principle. Next, the Galerkin method was applied to discretize the motion equations and the governing equations were simplified into a 1DOF nonlinear vibration differential equation under Volmir’s assumption. Finally, the nonlinear motion equations were solved with the harmonic balance method and the Runge-Kutta method, and the amplitude frequency response characteristic curves of the FGMs conical shells were obtained. The effects of different material distribution functions and different ceramic volume fraction exponents on the amplitude frequency response curves of conical shells were discussed. The bifurcation diagrams of conical shells with different cone angles, as well as time process diagrams and phase diagrams for different excitation amplitudes, were described. The motion characteristics were characterized by Poincaré maps. The results show that, the FGMs conical shells present the nonlinear characteristics of hardening springs. The chaotic motions of the FGMs conical shells are restrained and not prone to motion instability with the increase of the cone angle. The FGMs conical shell present a process from the periodic motion to the multi-periodic motion and then to chaos with the increase of the excitation amplitude. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Weak Graph Map Homotopy and Its Applications∗

The authors introduce a notion of a weak graph map homotopy (they call it M-homotopy), discuss its properties and applications. They prove that the weak graph map homotopy equivalence between graphs coincides with the graph homotopy equivalence defined by Yau et al in 2001. The difference between them is that the weak graph map homotopy transformation is defined in terms of maps, while the graph homotopy transformation is defined by means of combinatorial operations. They discuss its advantages over the graph homotopy transformation. As its applications, they investigate the mapping class group of a graph and the 1-order MP-homotopy group of a pointed simple graph. Moreover, they show that the 1-order MP-homotopy group of a pointed simple graph is invariant up to the weak graph map homotopy equivalence.