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1. Time-resolved velocity and ion sound speed measurements from simultaneous bow shock imaging and inductive probe measurements

   https://doi.org/10.1063/5.0098823  

   Datta, R.; Russell, D. R.; Tang, I.; Clayson, T.; Suttle, L. G.; Chittenden, J. P.; Lebedev, S. V.; Hare, J. D. (October 2022, Review of Scientific Instruments)

   We present a technique to measure the time-resolved velocity and ion sound speed in magnetized, supersonic high-energy-density plasmas. We place an inductive (“b-dot”) probe in a supersonic pulsed-power-driven plasma flow and measure the magnetic field advected by the plasma. As the magnetic Reynolds number is large ( R M > 10), the plasma flow advects a magnetic field proportional to the current at the load. This enables us to estimate the flow velocity as a function of time from the delay between the current at the load and the signal at the probe. The supersonic flow also generates a hydrodynamic bow shock around the probe, the structure of which depends on the upstream sonic Mach number. By imaging the shock around the probe with a Mach–Zehnder interferometer, we determine the upstream Mach number from the shock Mach angle, which we then use to determine the ion sound speed from the known upstream velocity. We use the sound speed to infer the value of [Formula: see text], where [Formula: see text] is the average ionization and T e is the electron temperature. We use this diagnostic to measure the time-resolved velocity and sound speed of a supersonic ( M S ∼ 8), super-Alfvénic ( M A ∼ 2) aluminum plasma generated during the ablation stage of an exploding wire array on the Magpie generator (1.4 MA, 250 ns). The velocity and [Formula: see text] measurements agree well with the optical Thompson scattering measurements reported in the literature and with 3D resistive magnetohydrodynamic simulations in GORGON. 

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2. Load Balancing in Parallel Queues and Rank-Based Diffusions

   https://doi.org/10.1287/moor.2023.0050  

   Banerjee, Sayan; Budhiraja, Amarjit; Estevez, Benjamin (May 2025, Mathematics of Operations Research)

   Consider a queuing system with K parallel queues in which the server for each queue processes jobs at rate n and the total arrival rate to the system is [Formula: see text], where [Formula: see text] and n is large. Interarrival and service times are taken to be independent and exponentially distributed. It is well known that the join-the-shortest-queue (JSQ) policy has many desirable load-balancing properties. In particular, in comparison with uniformly at random routing, the time asymptotic total queue-length of a JSQ system, in the heavy traffic limit, is reduced by a factor of K. However, this decrease in total queue-length comes at the price of a high communication cost of order [Formula: see text] because at each arrival instant, the state of the full K-dimensional system needs to be queried. In view of this, it is of interest to study alternative routing policies that have lower communication costs and yet have similar load-balancing properties as JSQ. In this work, we study a family of such rank-based routing policies, which we will call Marginal Size Bias Load-Balancing policies, in which [Formula: see text] of the incoming jobs are routed to servers with probabilities depending on their ranked queue length and the remaining jobs are routed uniformly at random. A particular case of such routing schemes, referred to as the marginal JSQ (MJSQ) policy, is one in which all the [Formula: see text] jobs are routed using the JSQ policy. Our first result provides a heavy traffic approximation theorem for such queuing systems in terms of reflected diffusions in the positive orthant [Formula: see text]. It turns out that, unlike the JSQ system, where, due to a state space collapse, the heavy traffic limit is characterized by a one-dimensional reflected Brownian motion, in the setting of MJSQ (and for the more general rank-based routing schemes), there is no state space collapse, and one obtains a novel diffusion limit which is the constrained analogue of the well-studied Atlas model (and other rank-based diffusions) that arise from certain problems in mathematical finance. Next, we prove an interchange of limits ([Formula: see text] and [Formula: see text]) result which shows that, under conditions, the steady state of the queuing system is well approximated by that of the limiting diffusion. It turns out that the latter steady state can be given explicitly in terms of product laws of Exponential random variables. Using these explicit formulae, and the interchange of limits result, we compute the time asymptotic total queue-length in the heavy traffic limit for the MJSQ system. We find the striking result that, although in going from JSQ to MJSQ, the communication cost is reduced by a factor of [Formula: see text], the steady-state heavy traffic total queue-length increases by at most a constant factor (independent of n, K) which can be made arbitrarily close to one by increasing a MJSQ parameter. We also study the case where the system is overloaded—namely, [Formula: see text]. For this case, we show that although the K-dimensional MJSQ system is unstable, unlike the setting of random routing, the system has certain desirable and quantifiable load-balancing properties. In particular, by establishing a suitable interchange of limits result, we show that the steady-state difference between the maximum and the minimum queue lengths stays bounded in probability (in the heavy traffic parameter n). Funding: Financial support from the National Science Foundation [RTG Award DMS-2134107] is gratefully acknowledged. S. Banerjee received financial support from the National Science Foundation [NSF-CAREER Award DMS-2141621]. A. Budhiraja received financial support from the National Science Foundation [Grant DMS-2152577]. 

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3. Linear instability of Z-pinch in plasma: Inviscid case

   https://doi.org/10.1142/S0218202521500093  

   Bian, Dongfen; Guo, Yan; Tice, Ian (February 2021, Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   The [Formula: see text]-pinch is a classical steady state for the MHD model, where a confined plasma fluid is separated by vacuum, in the presence of a magnetic field which is generated by a prescribed current along the [Formula: see text] direction. We develop a variational framework to study its stability in the absence of viscosity effect, and demonstrate for the first time that such a [Formula: see text]-pinch is always unstable. Moreover, we discover a sufficient condition such that the eigenvalues can be unbounded, which leads to ill-posedness of the linearized MHD system. 

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4. A MATHEMATICAL STUDY OF THE IMPLICIT ROLE OF INNATE AND ADAPTIVE IMMUNE RESPONSES ON WITHIN-HUMAN PLASMODIUM FALCIPARUM PARASITE LEVELS

   https://doi.org/10.1142/S0218339020400069  

   NGWA, GIDEON A.; WOLDEGERIMA, WOLDEGEBRIEL A.; TEBOH-EWUNGKEM, MIRANDA I. (June 2020, Journal of Biological Systems)

   A within-human-host malaria parasite model, integrating key variables that influence parasite evolution-progression-advancement, under innate and adaptive immune responses, is analyzed. The implicit role of immunity on the steady state parasite loads and parasitemia reproduction number ([Formula: see text]), a threshold parameter measuring the parasite’s annexing ability of healthy red blood cells (HRBCs), eventually rendering a human infectious to mosquitoes, is investigated. The impact of the type of recruitment function used to model HRBC growth is also investigated. The model steady states and [Formula: see text], both obtained as functions of immune system variables, are analyzed at snapshots of immune sizes. Model results indicate that the more the immune cells, innate and adaptive, the more efficient they are at inhibiting parasite development and progression; consequently, the less severe the malaria disease in a patient. Our analysis also illustrates the existence of a Hopf bifurcation leading to a limit cycle, observable only for the nonlinear recruitment functions, at reasonably large [Formula: see text]. 

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5. On Φ-variation for 1-d scalar conservation laws

   https://doi.org/10.1142/S0219891620500277  

   Jenssen, Helge Kristian; Ridder, Johanna (December 2020, Journal of Hyperbolic Differential Equations)

   null (Ed.)

   Let [Formula: see text] be a convex function satisfying [Formula: see text], [Formula: see text] for [Formula: see text], and [Formula: see text]. Consider the unique entropy admissible (i.e. Kružkov) solution [Formula: see text] of the scalar, 1-d Cauchy problem [Formula: see text], [Formula: see text]. For compactly supported data [Formula: see text] with bounded [Formula: see text]-variation, we realize the solution [Formula: see text] as a limit of front-tracking approximations and show that the [Formula: see text]-variation of (the right continuous version of) [Formula: see text] is non-increasing in time. We also establish the natural time-continuity estimate [Formula: see text] for [Formula: see text], where [Formula: see text] depends on [Formula: see text]. Finally, according to a theorem of Goffman–Moran–Waterman, any regulated function of compact support has bounded [Formula: see text]-variation for some [Formula: see text]. As a corollary we thus have: if [Formula: see text] is a regulated function, so is [Formula: see text] for all [Formula: see text]. 

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