• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1305-1315
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• 2022

Let u be a function on a connected finite graph G = (V, E). We consider the mean field equation (1) $-{\Delta}u={\rho}\({\frac{he^u}{\int_Vhe^ud{\mu}}}-{\frac{1}{{\mid}V{\mid}}}\),$ where ∆ is 𝜇-Laplacian on the graph, 𝜌 ∈ ℝ\{0}, h : V → ℝ⁺ is a function satisfying min_x∈V h(x) > 0. Following Sun and Wang [15], we use the method of Brouwer degree to prove the existence of solutions to the mean field equation (1). Firstly, we prove the compactness result and conclude that every solution to the equation (1) is uniformly bounded. Then the Brouwer degree can be well defined. Secondly, we calculate the Brouwer degree for the equation (1), say $$d_{{\rho},h}=\{{-1,\;{\rho}>0, \atop 1,\;{\rho}<0.}$$ Consequently, the equation (1) has at least one solution due to the Brouwer degree d_𝜌,h ≠ 0.

• Journal of KIISE:Information Networking
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• v.30 no.4
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• pp.545-558
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• 2003

It is expected that per-user customized services are widely used in next generation Personal Communication Network. The ultimate goal for personalized service is the Virtual home Environment (VHE) providing ´same-look-and-feel´ services for the subscriber wherever he roams to. To provide personalized services for each call, per-user service profiles are frequently referenced, so efficient service profile management is essentially required. To realized the VHE, typically two schemes, can be employed; One is Intelligent Network based service control and the other is a full replication scheme that always replicates profile in user´s current zone. The first scheme is referred as Central scheme and th second scheme is the modified replication scheme of IMT-2000, we refer to as Follow-Me Replication Unconditional (FMRU). Since the Central scheme only depends on the service cal rate and the FMRU is merely dependent on the movement rate, it is apparent that FMRU scheme outperforms the Central scheme if the call to mobility ratio (CMR) is large, and vice versa. In this paper, we propose a new service profile replication schemes, Adaptive Follow-Me Replication (AFMR) that determine replication automatically according to the user´s CMR. We compared the performance of the AFMR with the non-adaptive Follow-Me Replication unconditional on Demand (FMRUD) scheme. Performance results indicate that as the CMR of a user changes AFMR adapts well compared to the existing schemes.