The behavior of differential forms under purely inseparable extensions

Let F be a field of characteristic 2. In this paper we give a complete computation of the kernel of the homomorphism $H_2^{m+1}(F)?H_2^{m+1}(L)$ induced by scalar extension, where $L/F$ is a purely inseparable extension (of any degree), $H_2^{m+1}(F)$ is the cokernel of the Artin–Schreier operator $?:{\mathrm{\Omega }}_F^m?{\mathrm{\Omega }}_F^m/\mathrm{d}{\mathrm{\Omega }}_F^{m?1}$ given by: $x\frac{\mathrm{d}{x}_1}{x_1}\wedge ?\wedge \frac{\mathrm{d}{x}_m}{x_m}?(x^2?x)\frac{\mathrm{d}{x}_1}{x_1}\wedge ?\wedge \frac{\mathrm{d}{x}_m}{x_m}+\mathrm{d}{\mathrm{\Omega }}_F^{m?1}$, where ${\mathrm{\Omega }}_F^m$ is the space of absolute m-differential forms over F and d is the differential operator. Other related results are included.     

Synthesis and characterization of indium and thallium immobilized on isonicotinamide‐functionalized mesoporous MCM‐41: Two novel and highly active heterogeneous catalysts for selective oxidation of sulfides and thiols to their corresponding sulfoxides and disulfides

Two highly ordered isonicotinamide (INA)‐functionalized mesoporous MCM‐41 materials supporting indium and thallium (MCM‐41‐INA‐In and MCM‐41‐INA‐Tl) have been developed using a covalent grafting method. A surface functionalization method has been applied to prepare Cl‐modified mesoporous MCM‐41 material. Condensation of this Cl‐functionalized MCM‐41 with INA leads to the formation of MCM‐41‐INA. The reaction of MCM‐41‐INA with In(NO₃)₃ or Tl(NO₃)₃ leads to the formation of MCM‐41‐INA‐In and MCM‐41‐INA‐Tl catalysts. The resulting materials were characterized using various techniques. These MCM‐41‐INA‐In and MCM‐41‐INA‐Tl catalysts show excellent catalytic performance in the selective oxidation of sulfides and thiols to their corresponding sulfoxides and disulfides. Finally, it is found that the anchored indium and thallium do not leach out from the surface of the mesoporous catalysts during reaction and the catalysts can be reused for seven repeat reaction runs without considerable loss of catalytic performance.     

Perturbative manifolds and the Noether generators of nth-order Poisson equations

A manifold that contains small perturbations will induce a perturbed partial differential equation. The partial differential equation that we select is the Poisson equation – in order to explore the interplay between the geometry of the manifold and the perturbations. Specifically, we show how the problem of symmetry determination, for higher-order perturbations, can be elegantly expressed via geometric conditions.     

Two-dimensional vortex sheets for the nonisentropic Euler equations: Nonlinear stability

We show the short-time existence and nonlinear stability of vortex sheets for the nonisentropic compressible Euler equations in two spatial dimensions, based on the weakly linear stability result of Morando and Trebeschi (2008) [20]. The missing normal derivatives are compensated through the equations of the linearized vorticity and entropy when deriving higher-order energy estimates. The proof of the resolution for this nonlinear problem follows from certain a priori tame estimates on the effective linear problem in the usual Sobolev spaces and a suitable Nash–Moser iteration scheme.     

Path integral quantization corresponding to the deformed Heisenberg algebra

In this paper, the deformation of the Heisenberg algebra, consistent with both the generalized uncertainty principle and doubly special relativity, has been analyzed. It has been observed that, though this algebra can give rise to fractional derivative terms in the corresponding quantum mechanical Hamiltonian, a formal meaning can be given to them by using the theory of harmonic extensions of function. Depending on this argument, the expression of the propagator of the path integral corresponding to the deformed Heisenberg algebra, has been obtained. In particular, the consistent expression of the one dimensional free particle propagator has been evaluated explicitly. With this propagator in hand, it has been shown that, even in free particle case, normal generalized uncertainty principle and doubly special relativity show very much different result.     

Recent Progress in the $L_p$ Theory for Elliptic and Parabolic Equations with Discontinuous Coefficients

In this paper, we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients. We begin with an approach given by N. V. Krylov to parabolic equations in the whole space with $\rm{VMO}_x$ coefficients. We then discuss some subsequent development including elliptic and parabolic equations with coefficients which are allowed to be merely measurable in one or two space directions, weighted $L_p$estimates with Muckenhoupt ($A_p$) weights, non-local elliptic and parabolic equations, as well as fully nonlinear elliptic and parabolic equations.     

Cycloparaphenylene (CPP) series are molecular models for graphene armchair edges: trends in their aromaticity and cyclic conjugation are evaluated

Cycloparaphenylene ([r]CPP) and cyclacene ([r]CA) series are models for short carbon nanotubes. It is shown that armchair edges in model cycloparaphenylenes possess greater aromaticity and cyclic conjugation than do zigzag edges in model cyclacenes. According to Aihara’s bond resonance energy (BRE) and Bosanac and Gutman energy effect (ef) measurements, cycloparaphenylenes are twice as aromatic as cyclacenes. The general solution of all eigenvalues of all members of the cycloparaphenylene series is given. The origin of the recurrence of some eigenvalues are determined.     

Gas-phase 21Ne NMR studies and the nuclear magnetic dipole moment of neon-21

Gas-phase ²¹Ne nuclear magnetic resonance spectra were measured at the natural abundance of ²¹Ne isotope for samples consisting of pressurized neon up to 60 bar at room temperature and applying the magnetic field of the strength B₀ = 11.7574 T. It showed that the nuclear magnetic resonance frequency is linearly dependent on the density of gaseous neon. The resonance frequency was extrapolated to the zero-density point, and it permitted the determination of the ²¹Ne nuclear magnetic moment, μ(²¹Ne) = 0.6617774(10) μ_N. The present value of μ(²¹Ne) is not influenced by the bulk magnetic susceptibility of neon and interactions between neon atoms; therefore, it is more precise and reliable than the previous result obtained for μ(²¹Ne).