Essential maps and coincidence theory

Using an elementary approach, we present new coincidence principles for general classes of maps.     

Degrees-Of-Freedom in Multi-Cloud Based Sectored Cellular Networks

This paper investigates the achievable per-user degrees-of-freedom (DoF) in multi-cloud based sectored hexagonal cellular networks (M-CRAN) at uplink. The network consists of N base stations (BS) and K≤N base band unit pools (BBUP), which function as independent cloud centers. The communication between BSs and BBUPs occurs by means of finite-capacity fronthaul links of capacities CF=μF·12log(1+P) with P denoting transmit power. In the system model, BBUPs have limited processing capacity CBBU=μBBU·12log(1+P). We propose two different achievability schemes based on dividing the network into non-interfering parallelogram and hexagonal clusters, respectively. The minimum number of users in a cluster is determined by the ratio of BBUPs to BSs, r=K/N. Both of the parallelogram and hexagonal schemes are based on practically implementable beamforming and adapt the way of forming clusters to the sectorization of the cells. Proposed coding schemes improve the sum-rate over naive approaches that ignore cell sectorization, both at finite signal-to-noise ratio (SNR) and in the high-SNR limit. We derive a lower bound on per-user DoF which is a function of μBBU, μF, and r. We show that cut-set bound are attained for several cases, the achievability gap between lower and cut-set bounds decreases with the inverse of BBUP-BS ratio 1r for μF≤2M irrespective of μBBU, and that per-user DoF achieved through hexagonal clustering can not exceed the per-user DoF of parallelogram clustering for any value of μBBU and r as long as μF≤2M. Since the achievability gap decreases with inverse of the BBUP-BS ratio for small and moderate fronthaul capacities, the cut-set bound is almost achieved even for small cluster sizes for this range of fronthaul capacities. For higher fronthaul capacities, the achievability gap is not always tight but decreases with processing capacity. However, the cut-set bound, e.g., at 5M6, can be achieved with a moderate clustering size.     

A note on reactive equilibria in the discounted prisoner's dilemma and associated games

Necessary and sufficient conditions are derived under which the Tit-for-Tat strategy pair is a subgame perfect equilibrium in the standard discounted prisoner's dilemma. The sensitivity of the equilibrium to changes in the discount parameters is shown to be a common feature of all such nontrivial “reactive” equilibria. This appears to rule them out as reasonable solutions to the discounted prisoner's dilemma. In remarks, we indicate how these results can be extended to a fairly general class of games.     

An Extension of Polyak’s Theorem in a Hilbert Space

Let H be an infinite-dimensional real Hilbert space equipped with the scalar product (⋅,⋅) _H . Let us consider three linear bounded operators,

A study of fractional Lotka-Volterra population model using Haar wavelet and Adams-Bashforth-Moulton methods

The Lotka-Volterra (LV) system is an interesting mathematical model because of its significant and wide applications in biological sciences and ecology. A fractional LV model in the Caputo sense is investigated in this paper. Namely, we provide a comparative study of the considered model using Haar wavelet and Adams-Bashforth-Moulton methods. For the first method, the Haar wavelet operational matrix of the fractional order integration is derived and used to solve the fractional LV model. The main characteristic of the operational method is to convert the considered model into an algebraic equation which is easy to solve. To demonstrate the efficiency and accuracy of the proposed methods, some numerical tests are provided.