The relative importance of temperature and isotope effects on the dipole oscillator strength distribution of H2

Summary The calculation of the effects of temperature and isotopic composition on the energy weighted moments of the dipole oscillator strength distribution of H₂ in the random phase approximation to the polarization propagator are reported. It is seen that the effect of isotopic composition is small, while that of temperature is of an order accessible to experiment. We find that all the mean excitation energiesI , for =–1, 0, 1, decrease with temperature as does the dipole oscillator strength momentS() for >0, while the opposite is true for <0. These effects are interpreted in terms of the bond length dependence of the excitation energies.     

Periodic solutions of dry friction problems

Dry friction problems lead to discontinuous differential equations, e.g. to

The complexity of a class of infinite graphs

IfG _k is the family of countable graphs with nok vertex (or edge) disjoint circuits (1<k<) then there is a countableG _k G _k such that every member ofG _k is an (induced) subgraph of some member ofG _k , but no finiteG _k suffices.     

A constructive method for uniform approximation by means of integrated Lagrange interpolation polynomials in the spaceC 1[a,b]

Пусть интерполяционные уз лыa <x ₁ <x ₂ < ... <х _n <b являются корнямиn-го полинома системы полиномов {w _n(x)} _n ^∞ =1, ортогонально й с весомρ(х), 0 < ρ(х) ∈L[a, b], причем 1/ρ(x) ∈L[a, b]. Доказано, что еслиf(х) ∈С ¹ [a, b], то проинтегрированн ые полиномы $$Q_n (x) = \smallint _a^x L_{n - 1} (f',t)dt + f(a)$$ интерполяционных по линомов Лагранжа $$L_{n - 1} (f,x) = \sum\limits_{k = 1}^n {f'(x_k )\frac{{\omega _n (x)}}{{\omega '_n (x_k )(x - x_k )}}}$$ порядкаn?1 функцииf′(x) сходятся равномерно к функцииf(х) в замкнуто м интервале [а,b]. Во второй части стать и, где вместо условия 1/ρ(х) ∈L[a, b] предполагается, чтоρ(х) ∈С ²[а, b],ρ(х) > 0,ρ′ (х) > 0, доказываются подобные утверждени я для проинтегрированных интерполяционных по линомов Лагранжа и для прибли жений рациональными дробя ми. В работе даются неулу чшаемые порядковые о ценки и описываются примене ния полученных результа тов в теории дифферен циальных уравнений.     

Filtering on manifolds

We coasider a partially observable diffusion process (x _t,y_t)t0 whose unobservable componentx _t lives on a submanifold M ofR ⁿ . We present some general conditions under which the conditional law ofx _t, given the observationsy _s ,s [0,t], admits a density w.r.t. a given measure on M. We characterize the analytical properties of this density by using appropriate Sobolev spaces.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.     

Fractal dimension,attractors, and the boussinesq approximation in three dimensions

The Boussinesq approximation, where the viscosity depends polynomially on the shear rate, finds more and more frequent use in geological practice. In the paper, this modified Boussinesq approximation is investigated as a dynamical system for which the existence of a global attractor is proved. Finally, a new criterion for estimating the fractal dimension of invariant sets is formulated and its application to the problem under consideration is illustrated.     

On approximation by bivariate incomplete polynomials

In this paper we prove that given certain convex domains Δ on the plane, ε>0, andf∈C(Δ) such thatf=0 on θ²Δ={(θ² x,θ² y):(x,y)?Δ} (0<θ<1), a polynomialp(x, y) of the form $$p(x,y) = \sum\limits_{\theta n \leqslant k + l \leqslant n} {a_{kl} x^k y^l }$$ exists such that ∥f?p∥ _C(Δ) ≤ε. The admissible convex domains include triangles and parallelograms with a vertex at the origin and sections of unit disk.     

The Green relations approach to congruences on completely regular semigroups

On the congruence lattice (2(5) of a completely regular semigroup S the following mappings are considered _p: P and _p: VP, whereP is any of the Green relationsH, L, R orD. The equivalence relationsP andP ^V induced by these maps represent the main object of study in the paper. The former is a complete -congruence whereas the latter is a complete congruence onC(S). In particularH ,H ^V,L ^V,R ^V coincide with the kernel, trace, left trace and right trace relations onC(S), respectively. All essential properties known for the latter relations carry over to the new relationsP andP ^V. In addition, some interesting interplays of these provide for more richness in the theory of congruences on completely regular than is the case for the kernel-trace approach to congruences on regular semigroups.     

Homogenization of the heat equation for a domain with a network of pipes with a well-mixed fluid

We consider the linear heat equation in a domain occupied by a solid material with a network of pipes in which a well-mixed fluid is circulating. The temperature of the fluid in the pipe is uniform and its time variation is determined by the thermal flux on the wall of the pipe, plus a given internal source; continuity of the temperature across the pipe is also assumed. We suppose that we deal with a periodic geometry, with cells of size with inclusions of size r_g; we study in detail in the case r, referring to a previous paper for the case r In the limit »0 we get a homogenized equation. The limit depends strongly on the ratio between the time variation of the temperature in the inclusions and the thermal flux through the interface. The homogenized equation has a new specific heat, which depends on the porosity and the constant of proportionality between the time variation of temperature and the flux on the boundary of the pipe. We also have a new thermal conductivity depending on the microstructure, and volume sources appear. The main tool is the energy method and we generalize the classical results for the more standard boundary conditions for parabolic equations. Finally, we consider the network of pipes forming a random ball structure. We prove convergence for this case. The homogenized equation is of the same form as in the periodic case but auxiliary problems are stochastic.