Given a function v ∈ BV (Ω; R m ), we introduce the notion of a minimal lifting of Dv. We prove that every v ∈ BV (Ω; R m ) has a unique minimal lifting, and we show that if v k → v strictly in BV , then the minimal liftings of v k converge weakly as measures to the minimal lifting of v. As an application, we deduce a result about weak continuity of the distributional determinant Det D 2 u with respect to strict convergence.

We consider the pandemic spread of COVID-19 in selected countries after the outbreak of the SARS-CoV-2 coronavirus in Wuhan City, China. We estimated the infection rate and the initial individuals infected with COVID-19 by using officially reported data from the early stages of the epidemic for a model of susceptible (S), infectible (I), quarantined (Q), and officially confirmed recovered (R k) populations (the so-called SIQR k model). In the officially reported data, we know the number of quarantined cases and the officially reported number of recovered cases. We cannot know about recovered cases from asymptomatic patients. In the SIQR k model, we can estimate the parameters and the initial infections (confirmed cases + asymptomatic cases) from fitted values. We obtained an infection rate in the range β = 0.233 ∼ 0.462, a basic reproduction number of R o = 1.8 ∼ 3.5, and the initial number of infected individuals, I (0) = 10 ∼ 8409, for selected countries. By using fitting parameters, we estimated that the maximum time span of the infection was around 50 days in Germany when the government invoked the quarantine policy. The disease is expected to subside about 6 months after the first patients are found.

We review the agent-based models (ABM) on social physics including econophysics. The ABM consists of agent, system space, and external environment. The agent is autonomous and decides his/her behavior by interacting with the neighbors or the external environment with the rules of behavior. Agents are irrational because they have only limited information when they make decisions.They adapt using learning from past memories. Agents have various attributes and are heterogeneous. ABM is a non-equilibrium complex system that exhibits various emergence phenomena. The social complexity ABM describes human behavioral characteristics. In ABMs of econophysics, we introduce the Sugarscape model and the artificial market models. We review minority games and majority games in ABMs of game theory. Social flow ABM introduces crowding, evacuation, traffic congestion, and pedestrian dynamics. We also review ABM for opinion dynamics and voter model. We discuss features and advantages and disadvantages of Netlogo, Repast, Swarm, and Mason, which are representative platforms for implementing ABM.

In this paper, we study differential equations arising from the generating function of the (r, β)-Bell polynomials. We give explicit identities for the (r, β)-Bell polynomials. Finally, we find the zeros of the (r, β)-Bell equations with numerical experiments.

In this paper we investigate some symmetric property of the twisted q-Euler zeta functions and twisted q-Euler polynomials.