Notion Of Uniform Convergence Research Articles

Mathematical Modelling and Computational Dynamics of the impact of Quasi-Lockdown Policy in Control of COVID-19 in Akwa Ibom State, Nigeria

Determining the impact of a local control strategy, implemented to curtail the spread of COVID-19 has become obligatory to enable policy makers to combat the so-called long covid. This research used a deterministic SEIAHR-model, incorporating a logistic function to investigate the effect of quasi-lockdown policy during reemergence of corona virus pandemic. The notion of uniform convergence and fixed point theory were used to establish positive invariant region, ultimate boundedness, and existence of unique solution of the model. Nonlinear dynamical behaviours of the model such as stability and oscillating flows occurred at the disease free, and endemic equilibrium points. Using centre manifold theory, a trans-critical bifurcation with hysteresis effect were established, when quasi-lockdown policy was used as a control parameter in the model. A forward sensitivity analysis was conducted to unfold parameters that contributed significantly towards the spread and control of the infection. These parameters were estimated and fitted using least square technique, comparative to the observed real datasets adapted from the Nigeria Centre for Disease Control (NCDC). The polynomial regression model fitted the observed datasets with an average coefficient of determination given as R2 = 0.94. Simulations in form of phase space diagrams were used to validate the theoretic results. Epidemiologically, the model demonstrated that quasi-lockdown policy was necessarily a strategic policy to curb the spread of the infection, but not sufficient to eradicate the disease as the disease persisted endemically. It was shown that the basic reproduction number of the infection was within the critical threshold (R0 \(\le\) 1), when the lockdown policy was enforced strictly, but otherwise on relaxation of the lockdown policy (R0 \(\ge\) 1) Thus a relaxation of the quasi-lockdown policy leads to increase in susceptibility of the populace as the infection risk ratio increases in the model. Similarly, the model predicted that a pre-mature lifting of the lockdown policy could have led to high infectivity on the susceptible class and disastrous to healthy living of the citizens.

Uniform convergence of a sequence of functions at a point

It is well known that the uniform limit of a sequence of continuous real-valued functions defined on an interval I is itself continuous. However, if the convergence is pointwise, the limit function need not be continuous (take ƒ n (x) = x n on [0, 1], for example). Boas has shown that the pointwise limit function of a sequence of continuous real-valued functions defined on the compact interval [a,b] is, nonetheless, continuous on a dense subset of [a,b]. In this paper, the notion of uniform convergence at a point is offered as an alternative to the Boas approach in establishing this and, consequently, other results. The arguments stay within the realm of a first proof course in classical mathematical analysis.

On the expansion of a function in terms of spherical harmonics in arbitrary dimensions

It is well known that a function which is continuously differentiable on the unit circle S can be expressed as a uniformly convergent Fourier series. It is less known that a function which is continuously differentiable on the unit sphere S in R can be expanded in terms of a uniformly convergent series of spherical harmonics, a so–called Laplace series 1 (Kellogg [20,p.259]). Both results can be traced back at least to Dirichlet (1829 and 1837, respectively), although the notion of uniform convergence was brought out a little later through the work of Gudermann (the teacher of Weierstrass), Seidel (a student of Dirichlet), Stokes and Weierstrass. The importance of such an expansion is due to the fact that the solution of the Dirichlet problem for the Laplace equation on the unit disc or unit ball can then be given in terms of a uniformly convergent series of elementary functions. The first edition of Heine’s handbook of spherical harmonics which appeared in 1861 and reproduced Dirichlet’s 1837 proof does not contain the notion of uniform convergence. The second edition of 1878 does have this notion [14,p.478f.]; at the same time it takes a critical attitude towards Dirichlet’s proof [14,p.434]. We shall comment on this proof in Remark 1 at the end of §2.

On uniform convergence of linear operators on the probabilistic normed space

In this paper, we introduced the notion of uniform convergence of the linear operators on the probabilistic normed space, and the notion of probabilistic distance between the operators, which describes the above convergence completely. In terms of these notions, we obtained the essential features of the continuity of operators, and of the uniform convergence of operator sequences, and we also obtained the closure of continuity and complete continuity under the operation of the limit of uniform convergence.

On the notion of uniform convergence with respect to a fundamental set of functionals, and its applications

On the notion of uniform convergence with respect to a fundamental set of functionals, and its applications

Dense imbedding of topological groups

The present article originated from the problem of determining when a continuous representation of a Lie group into another is open. Important cases of the problem have been discussed by A. Malcev2 and the author,3 and some of them have been extended to the case of more general groups by the author and H. Yamabe.4 Recently W. T. van Est obtained new results concerning the same problem on Lie groups.6 Here I shall give an extension of his essential result to a more general case in a simpler way. Let G be a locally compact connected group and let A (G) be the group of all continuous automorphisms of G. Let A (G) be topologized by the notion of uniform convergence in the wider sense. Now let I(G) be the subgroup of A (G) composed of all inner automorphisms of G. We shall call G a (CA) group6 if I(G) is a closed subgroup of A(G).