Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on S1

We investigate the exact-WKB analysis for quantum mechanics in a periodic potential, with N minima on S1. We describe the Stokes graphs of a general potential problem as a network of Airy-type or degenerate Weber-type building blocks, and provide a dictionary between the two. The two formulations are equivalent, but with their own pros and cons. Exact-WKB produces the quantization condition consistent with the known conjectures and mixed anomaly. The quantization condition for the case of N-minima on the circle factorizes over the Hilbert sub-spaces labeled by discrete theta angle (or Bloch momenta), and is consistent with ’t Hooft anomaly for even N and global inconsistency for odd N. By using Delabaere-Dillinger-Pham formula, we prove that the resurgent structure is closed in these Hilbert subspaces, built on discrete theta vacua, and by a transformation, this implies that fixed topological sectors (columns of resurgence triangle) are also closed under resurgence.

Midterm Outcomes of Unstable Ankle Fractures in Young Patients Treated by Closed Reduction and Fixation With an Intramedullary Fibular Nail vs Open Reduction Internal Fixation Using a Lateral Locking Plate

Background: We aimed to compare midterm radiological and clinical outcomes between closed reduction and internal fixation (CRIF) using the fibular intramedullary nail (IMN) and open reduction and internal fixation (ORIF) using the locking plate for the treatment of unstable ankle fractures in active young patients. Methods: In this retrospective cohort study, 204 patients treated with CRIF using the fibular IMN (94 patients) or ORIF using the locking plate (110 patients) were included after at least 3 years of follow-up. The mean patient age was 41.4 years. Radiographic evaluation included the quality of reduction assessed by plain radiography and 3-dimensional (3D)–reconstructed computed tomography as well as the development of posttraumatic osteoarthritis (PTOA) of the ankle assessed by weightbearing plain radiography. Clinical evaluation included the American Orthopaedic Foot & Ankle Society hindfoot score, Olerud and Molander Score, the Foot and Ankle Outcome Score, and visual analog scale pain score as well as complications. Results: At median follow-up greater than 4 years, we found no significant differences in measured clinical outcomes between the 2 groups. There were significantly fewer postoperative complications in the IMN group than in the ORIF group (9.5% vs 39%, P < .001). However, we did find a greater proportion of radiographically fair or poor reductions in the IMN group than in the ORIF group ( P < .001). The poor reductions in the IMN group were primarily related to Weber type C, pronation-type injury, and comminuted fibular and trimalleolar fractures ( P < .001). PTOA was also more frequently observed in the IMN group than in the ORIF group (21.3% vs 9.1%, P = .024). Conclusion: Given the current prevailing technologies for fracture fixation, this study suggests that surgeons should consider ORIF for unstable ankle fractures in active young patients with Weber type C, pronation-type injury, and comminuted fibular and trimalleolar fractures. Level of Evidence: Level III, retrospective comparative study.

N+ Irradiation and Substrate-Induced Variability in the Metamagnetic Phase Transition of FeRh Films

Metamagnetic FeRh has been the focus of numerous studies for its highly unique antiferromagnetic (AF) to ferromagnetic (FM) metamagnetic transition. While this phase transition usually occurs above room temperature (often Tc > 400 K), both ion irradiation and strained epitaxial growth have been used to bring it to applicable temperatures. Nevertheless, cross sample variability is pervasive in these studies. Here we explore the optical and magnetic properties of 35 nm thick FeRh grown by magnetron sputter deposition simultaneously on two different single crystal substrates: epitaxially on MgO (001) and highly strained with large lattice mismatch on Al2O3 (1000). We then irradiate the epitaxial film with 5 keV N+ ions to introduce disorder (and to a lesser extent, modify chemical composition) without effecting the surface morphology. We find that the phase-transitional properties of both films are strikingly different due to the large lattice mismatch, despite being grown in tandem with nominally identical growth conditions including Fe/Rh stoichiometry, pressure, and temperature. We observe that N+ implantation lowers Tc by ~60 K, yielding a sample with nominally the same transition temperature as the non-epitaxial film on sapphire, yet with a significantly increased magnetic moment, a larger magnetization change and a more abrupt transition profile. We attribute these differences to the Volmer-Weber type growth mode induced by the sapphire substrate and the resulting rougher surface morphology.

The Value of Additional Gravity Stress Radiographs for Decision Making in the Treatment of Isolated Type B Distal Fibular Fractures

Background: Prior to treatment decisions concerning isolated Weber type B ankle fractures, assessment of the stability of the ankle joint is mandatory. The gravity stress (GS) radiograph is a radiographic tool to determine stability. We hypothesized that this additional GS radiograph would lead to fewer operative treatments by applying the criterion of operative treatment when medial clear space (MCS) > superior clear space (SCS) + 2 mm on the GS radiograph, compared with the nonstressed mortise view criteria of advising operative treatment in case of MCS > SCS + 1 mm. Methods: This retrospective comparative cohort study analyzed 343 patients aged between 18 and 70 years with an isolated Weber type B ankle fracture diagnosed at the emergency department between January 2014 and December 2019. The cohort was divided into 2 groups based on whether an additional GS radiograph was performed. Group I consisted of 151 patients in whom a regular mortise and lateral radiograph were performed. Group II comprised 192 patients, with an additional GS radiograph. Primary outcome was type of treatment (conservative vs operative). Secondary outcomes were patient-reported functional outcomes and pain. Results: Baseline characteristics of both groups did not differ. In group I, surgery was performed in 60 patients (39.7%) compared with 108 patients (56.3%) in group II ( P = .002). In the operatively treated patients, the mean MCS on regular mortise view was significantly smaller in patients in whom an additional GS radiograph was performed compared to patients without an additional GS radiograph (4.1 mm vs 5.2 mm, P < .001). Mean Olerud-Molander Ankle Score and mean visual analog scale (VAS) for pain did not differ significantly between groups I and II. Conclusions: Contrary to what was hypothesized, the introduction of an additional gravity stress radiograph, by which operative treatment was indicated if the MCS was wider than the SCS + 2 mm, did not result in reduced operative treatment of Weber type B ankle fractures when operative treatment was indicated for MCS > SCS + 1 mm on non-gravity stress radiographs. Level of Evidence: Level III: retrospective comparative study.

Anterior distal tibial plafond-plasty for the treatment of posttraumatic ankle osteoarthritis with anterior translation of the talus

Posttraumatic ankle osteoarthritis (OA) represents a significant challenge to orthopedic surgeons, especially in cases of anterior talar translation and concomitant impaction of the anterior distal tibial plafond. The aim of this study was to evaluate the clinical outcomes of an intra-articular osteotomy for the management of these patients. A total of 21 patients meeting our criteria were retrospectively reviewed. Sixteen patients sustained initial pilon fractures, while five patients had Weber type C ankle fractures. Anterior distal tibial plafond-plasty was performed to address the impaction and anterior translation of the talus. The American Orthopedic Foot and Ankle Society (AOFAS) hindfoot score and visual analog scale (VAS) score were utilized as clinical outcomes. The lateral talar station (LTS), tibial lateral surface (TLS) angle, tibial anterior surface angle and talocrural angle were evaluated pre- and postoperatively. The modified Kellgren-Lawrence score was used for the evaluation of sagittal ankle OA. The average age at surgery was 35 years, and the average follow-up duration was 34 months. The AOFAS hindfoot score increased from 26 to 71 (p < 0.01), and the VAS score improved from 7 to 2 (p < 0.01). The LTS improved from 9.0 to 2.3 mm (p < 0.01), and the TLS angle improved from 72° to 81° (p < 0.01). Of the 21 patients, 18 showed improvement in or no worsening of ankle OA on the sagittal plane, while 3 developed advanced ankle OA. A congruent ankle joint on the sagittal plane could be achieved by anterior distal tibial plafond-plasty. This is a valuable treatment option for the salvage of posttraumatic ankle OA with anterior translation of the talus.

On Correlated Equilibria in Marinatto–Weber Type Quantum Games

Players’ choices in quantum game schemes are often correlated by a quantum state. This enables players to obtain payoffs that may not be achievable when classical pure or mixed strategies are used. On the other hand, players’ choices can be correlated due to a classical probability distribution, and if no player benefits by a unilateral deviation from the vector of recommended strategies, the probability distribution is a correlated equilibrium. The aim of this paper is to investigate relation between correlated equilibria and Nash equilibria in the MW-type schemes for quantum games.

Hybrid Fixation for Danis-Weber Type C Fractures With Syndesmosis Injury

Background: This study aimed to assess the clinical and radiological outcomes of hybrid fixation for Danis-Weber type C ankle fractures with a syndesmotic injury. Methods: From January 2016 to April 2018, we retrospectively reviewed consecutive patients who underwent hybrid fixation for Danis-Weber type C ankle fractures with syndesmotic injury with a minimum follow-up of 12 months. We excluded patients who achieved syndesmosis stability after fracture fixation. In all patients, we allowed partial weightbearing at 4 weeks postoperatively. We evaluated the visual analog scale for ankle pain, the Olerud-Molander ankle score, and the American Orthopaedic Foot & Ankle Society score. Malreduction of the syndesmosis was defined based on 2 previously reported methods. Fourteen patients (11 men and 3 women) were included in this study. There were 11 patients with pronation external rotation-type fractures and 3 patients with Maisonneuve fractures. The average age at operation was 37.2 years (range, 18-70 years). Results: Clinical scores were significantly improved at the last follow-up. Postoperative malreduction was observed only in 1 patient (7.1%). Conclusion: Hybrid fixation using a suture-button device combined with a syndesmotic screw in Danis-Weber type C fractures with syndesmosis injury showed a high accuracy of reduction, a low rate of diastasis, and favorable clinical outcomes. This combined method could be a good alternative treatment option for Danis-Weber type C ankle fractures with a syndesmosis injury. Level of Evidence: Level IV, case series.

Minimally invasive plate osteosynthesis (MIPO) versus open reduction and internal fixation (ORIF) in the treatment of distal fibula Danis-Weber types B and C fractures

Background Minimally invasive plate osteosynthesis (MIPO) has been reported to be superior to open reduction and internal fixation (ORIF) in the treatment of different long bone fractures. Nevertheless, in distal fibula fractures, the evidence of MIPO remains scarce. The aim of this retrospective study was to compare the clinical and radiological outcomes of the minimally invasive techniques applied to the distal fibula with open reduction and internal fixation within a 12 months follow-up. Methods A consecutive series of patients who underwent surgery using either ORIF or MIPO for the treatment of distal fibula fractures between 2010 and 2014 were retrospectively analyzed. All distal fibular fractures requiring an operative treatment (Danis-Weber type B ≙ AO type 44 B1, 2, 3 and Danis-Weber type C ≙ AO type 44 C1, 2) were included (ORIF n = 35, MIPO n = 35). Patients were assessed for postoperative pain using a visual analog scale (VAS) for pain (ranging from 0 to 10) and classified into 4 groups: “no pain” for VAS = 0, “low” for VAS = 1–3, “moderate” for VAS = 3–5, and “severe” for VAS = 5–10. In addition, complications of postoperative fracture-related infection, wound healing disorders, vascular and nerve injury and development of nonunion were evaluated and analyzed. Radiologic outcome measures assessing the talocrural angle, lateral and medial clear space, tibiofibular overlap, and talar tilt angle were evaluated postoperatively. Results The overall complication rate showed to be lower in the MIPO group compared to the ORIF group (14% vs. 37%, p = 0.029). Even though not statistically significant, specific surgery-related complications such as skin necrosis (3% vs. 9%, p = 0.275), nonunion (0% vs. 6%, p = 0.139), infections and wound healing disorders (9% vs. 20%, p = 0.141), as well as postoperative pain (17% vs. 26%, p = 0.5) were found more frequently in the ORIF group. The tibiofibular overlap demonstrated to be significantly lower in the ORIF group (3.3 mm vs. 2.7 mm, p = 0.033). The talocrural angle, talar tilt angle, and lateral and medial clear space showed to be equivalent in both groups. Conclusion In this retrospective single-center consecutive series, MIPO was superior to ORIF in the surgical treatment of distal fibula fractures with respect to the overall complication rate. Trial registration EKNZ Project-ID: 2019-02310, registered on the 20th of December 2019 with swissethics