Trade-off between job losses and the spread of COVID-19 in Japan

This paper quantitatively analyzes the trade-off between job losses and the spread of COVID-19 in Japan. We derive an empirical specification from the social planner’s resource constraint under the susceptible, infected, recovered, and deaths (SIRD) model and estimate how job losses and the case growth rate are related to people’s mobility using the Japanese prefecture-level panel data on confirmed cases, involuntary job losses, people’s mobility, and teleworkability. Our findings are summarized as follows. First, we find that a decrease in mobility driven by containment policies is associated with an increase in involuntary job separations, but the high teleworkability mitigates the negative effect of decreased mobility on job losses. Second, estimating how the case growth is related to people’s mobility and past cases, we find that the case growth rate is positively related to an increase in people’s mobility but negatively associated with past confirmed cases. Third, using these estimates, we provide a quantitative analysis of the trade-off between job losses and the number of confirmed cases. Taking Tokyo in July 2020 as a benchmark, we find that the cost of saving 1 job per month is 2.3 more confirmed cases per month in the short run of 1 month. When we consider a trade-off for 3 months from July to September of 2020, protecting 1 job per month requires 6.6 more confirmed cases per month. Therefore, the trade-off becomes worse substantially in the longer run of 3 months, reflecting the exponential case growth when the people’s mobility is high.

On two product form modifications for finite overflow systems

Overflow mechanisms can be found in a variety of queueing models. This paper studies a simple and generic overflow system that allows the service times to be both job type and station dependent. This system does not exhibit a product form. To justify simple product form computations, two product form modifications are given, as by a so-called call packing principle and by a stop protocol. The provided proofs are self-contained and straightforward for the exponential case and of merit by itself. Next, it is numerically studied whether and when, or under which conditions, the modifications lead to a reasonable approximation of the blocking probability, if not an ordering. The numerical results indicate that call packing provides a rather accurate approximation when the overflow station is not heavily utilized. Moreover, when overflowed jobs have an equal or faster service rate, the approximation is consistently found to be pessimistic, which can be useful for practical purposes. The stop protocol, in contrast, appears to be less accurate for most natural situations. Nevertheless, for an extreme situation the order might change. In addition, for the stop protocol the product form is proven to be insensitive (i.e. to also apply for arbitrary non-exponential service times). For call packing, this numerically appears not to be the case, as of interest by itself. However, from a practical viewpoint the sensitivity seems light. The results are intriguing for both theoretical and practical further research.

Insensitivity of the mean field limit of loss systems under SQ(d) routeing

In this paper, we study a large multi-server loss model under the SQ(d) routeing scheme when the service time distributions are general with finite mean. Previous works have addressed the exponential service time case when the number of servers goes to infinity, giving rise to a mean field model. The fixed point of the limiting mean field equations (MFEs) was seen to be insensitive to the service time distribution in simulations, but no proof was available. While insensitivity is well known for loss systems, the models, even with state-dependent inputs, belong to the class of linear Markov models. In the context of SQ(d) routeing, the resulting model belongs to the class of nonlinear Markov processes (processes whose generator itself depends on the distribution) for which traditional arguments do not directly apply. Showing insensitivity to the general service time distributions has thus remained an open problem. Obtaining the MFEs in this case poses a challenge due to the resulting Markov description of the system being in positive orthant as opposed to a finite chain in the exponential case. In this paper, we first obtain the MFEs and then show that the MFEs have a unique fixed point that coincides with the fixed point in the exponential case, thus establishing insensitivity. The approach is via a measure-valued Markov process representation and the martingale problem to establish the mean field limit.

Conjugate connections with respect to a quadratic endomorphism and duality

The goal of this paper is to consider the notion of conjugate connection in a unifying setting for both almost complex and almost product geometries, having as model the works of Mileva Prvanovic. A main interest is in finding classes of conjugate connections in duality with the initial linear connection; for example in the exponential case of almost complex geometry we arrive at a rule of quantization.

Remarks on Entire Solutions for Two Fourth-Order Elliptic Problems

We are interested in entire solutions for the semilinear biharmonic equation Δ2u = f(u) in ℝN, where f(u) = eu or –u–p (p > 0). For the exponential case, we prove that for the polyharmonic problem Δ2mu = eu with positive integer m, any classical entire solution verifies Δ2m–1u < 0; this completes the results of Dupaigne et al. (Arch. Ration. Mech. Analysis208 (2013), 725–752) and Wei and Xu (Math. Annalen313 (1999), 207–228). We also obtain a refined asymptotic expansion of the radial separatrix solution to Δ2u = eu in ℝ3, which answers a question posed by Berchio et al. (J. Diff. Eqns252 (2012), 2569–2616). For the negative power case, we show the non-existence of the classical entire solution for any 0 < p ⩽ 1.

Asymptotic Law of thejth Records in the Bivariate Exponential Case

We consider a sequence(Xi,Yi)1⩽i⩽nof independent and identically distributed random variables with joint cumulative distribution H(x,y), which has exponential marginalsF(x)andG(y)with parameterλ=1. We also assume thatXi(ω)≠Yi(ω),∀i∈N, andω∈Ω. We denoteRk(j)k⩾1andSk(j)k⩾1by the sequences of thejth records in the sequences(Xi)1⩽i⩽n,(Yi)1⩽i⩽n, respectively. The main result of of the paper is to prove the asymptotic independence ofRk(j)k⩾1andSk(j)k⩾1using the property of stopping time of thejth record times and that of the exponential distribution.