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*t*)= [sum ]

^{}

_{j=1}exp(-

*t*

_{j}) for small positive

*t*, where {

_{j}}

^{}

_{j=1}arethe eigenvalues of the negative Laplacian -

_{n}= -[sum ]

*(/*

^{n}_{k=1}*x*)

^{k}^{2}in R

*(*

^{n}*n*= 2 or 3), are studied for a general multiply connectedbounded domain which is surrounded by simply connected boundeddomains

_{i}with smooth boundaries

_{i}(

*i*= 1,...,

*m*), where smoothfunctions

*Y*(

_{i}*i*= 1,...,

*m*) are assuming the Robin boundary conditions(

*n*+

_{i}*Y*) = 0 on

_{i}_{i}. Here /

*n*denote differentiations along theinward-pointing normals to

_{i}_{i}(

*i*= 1,...,

*m*). Some applicationsof an ideal gas enclosed in the multiply connected bounded containerwith Neumann or Robin boundary conditions are given. 相似文献

*R*

^{2}with mixed boundary conditions, from the complete knowledge of the eigenvalues {

_{ j }}

_{ j=1}

^{}for the Laplace operator, using the asymptotic expansion of the spectral function (

*t*)=

_{ j=1}

^{}exp(–

*t*

_{ j }) as

*t*0. 相似文献

_{m}}

_{ m=1}

^{}are the eigenvalues of the Laplacian is studied for a variety of domains. The dependence of(

*t*) on the connectivity of a domain and the boundary conditions is analysed. Particular attention is given to annular domains.

Résumé Pour divers domaines, on étudie la fonction trace , où相似文献_{1},_{2},_{3}, sont les valeurs propres du laplacien. On analyse comment(t) dépend du domaine et des conditions aux limites. On considère notamment le cas de couronnes circulaires.

*t*) on the connectivity of domains and the boundary conditionsis analysed. Particular attention is given for a spherical shell. 相似文献

*ij*-normal, almost

*ij*-regular, quasi

*ij*-regular and strongly

*S*-

*ij*-regular. Using the introduced functions, we generalize several preservation theorems of normality and regularity to bitopological spaces. Implications between notions are given and counter examples for some reverse directions are obtained. It should be noted that considering the space time as the product of two topologies, the topology of space and that of the space time will open the way for new line of research in the field of quantum gravity initiated by Witten and El-Naschie and many others (cf. [Chaos, Solitons & Fractals 17 (2003) 989; Chaos, Solitons & Fractals 7 (1996) 499; Int. J. Theor. Phys. 37 (1998) 2935; Phys. Today (1996) 24]). 相似文献

^{16}O leaving sub-oxygen nucleus to break-up. The characteristics of these events of neutron emulsion collisions are displayed and compared with that obtained from incident proton at the same energy.

The studied interactions are classified according to the type of target nucleus (H, CNO and AgBr) by using the method of separation. Cascade-evaporation model (CEM) calculations are taken into account and compared with that obtained from the interactions of induced neutron from ^{16}O (4.5 A GeV/*c*) with emulsion nuclei and with the results of incident proton at the same target and energy. CEM can describe the distributions of proton and neutron interactions. It is conjuctured that it may be possible to develop a similar theoretical model based on the modern methods of nonlinear dynamics, neural network and complexity [Randomness and Undecidability in Physics, World Scientific, Singapore, 1993]. 相似文献