On Functions of quasi Toeplitz matrices

Let $a(z)=\sum_{i\in\mathbb Z}a_iz^i$ be a complex valued continuous function, defined for $|z|=1$, such that $\sum_{i=-\infty}^{+\infty}|ia_i|<\infty$. Consider the semi-infinite Toeplitz matrix $T(a)=(t_{i,j})_{i,j\in\mathbb Z^+}$ associated with the symbol $a(z)$ such that $t_{i,j}=a_{j-i}$. A quasi-Toeplitz matrix associated with the continuous symbol $a(z)$ is a matrix of the form $A=T(a)+E$ where $E=(e_{i,j})$, $\sum_{i,j\in\mathbb Z^+}|e_{i,j}|<\infty$, and is called a CQT-matrix. Given a function $f(x)$ and a CQT matrix $M$, we provide conditions under which $f(M)$ is well defined and is a CQT matrix. Moreover, we introduce a parametrization of CQT matrices and algorithms for the computation of $f(M)$. We treat the case where $f(x)$ is assigned in terms of power series and the case where $f(x)$ is defined in terms of a Cauchy integral. This analysis is applied also to finite matrices which can be written as the sum of a Toeplitz matrix and of a low rank correction

Efficient cyclic reduction for QBDs with rank structured blocks

We provide effective algorithms for solving block tridiagonal block Toeplitz systems with $m\times m$ quasiseparable blocks, as well as quadratic matrix equations with $m\times m$ quasiseparable coefficients, based on cyclic reduction and on the technology of rank-structured matrices. The algorithms rely on the exponential decay of the singular values of the off-diagonal submatrices generated by cyclic reduction. We provide a formal proof of this decay in the Markovian framework. The results of the numerical experiments that we report confirm a significant speed up over the general algorithms, already starting with the moderately small size $m\approx 10^2$

Equivalent Mid-Term Results of Open vs Endoscopic Gluteal Tendon Tear Repair Using Suture Anchors in Forty-Five Patients.

BackgroundLittle is known about the relative efficacy of open (OGR) vs endoscopic (EGR) gluteal tendon repair of gluteal tendon tears in minimizing pain and restoring function. Our aim is to compare these 2 surgical techniques and quantify their impact on clinical outcomes.MethodsAll patients undergoing gluteal tendon tear repair at our institution between 2015 and 2018 were retrospectively reviewed. Pain scores, limp, hip abduction strength, and the use of analgesics were recorded preoperatively and at last follow-up. The Hip disability and Osteoarthritis Outcome Score Junior and Harris Hip Score Section1 were obtained at last follow-up. Fatty degeneration was quantified using the Goutallier-Fuchs Classification (GFC). Statistical analysis was conducted using one-way analysis of variance and t-tests.ResultsForty-five patients (mean age 66, 87% females) met inclusion criteria. Average follow-up was 20.3 months. None of the 10 patients (22%) undergoing EGR had prior surgery. Of 35 patients (78%) undergoing OGR, 12 (27%) had prior hip replacement (75% via lateral approach). The OGRs had more patients with GFC ≥2 (50% vs 11%, P&nbsp;= .02) and used more anchors (P&nbsp;= .03). Both groups showed statistical improvement (P ≤ .01) for all outcomes measured. GFC &gt;2 was independently associated with a worst limp and Harris Hip Score Section 1 score (P&nbsp;= .05). EGR had a statistically higher opioid use reduction (P &lt; .05) than OGR. Other comparisons between EGR and OGR did not reach statistical significance.ConclusionIn this series, open vs endoscopic operative approach did not impact clinical outcomes. More complex tears were treated open and with more anchors. Fatty degeneration adversely impacted outcomes. Although further evaluation of the efficacy of EGR in complex tears is indicated, both approaches can be used successfully

Wearable activity sensors and early pain after total joint arthroplasty.

A prospective observational cohort of 20 primary total hip arthroplasty (n&nbsp;= 12) and total knee arthroplasty (n&nbsp;= 8) patients (mean age: 63 ± 6 years) was passively monitored with a consumer-level wearable activity sensor before and 6 weeks after surgery. Patients were clustered by minimal change or decreased activity using sensor data. Decreased postoperative activity was associated with greater pain reduction (-5.5 vs&nbsp;-2.0, P&nbsp;= .03). All patients surpassed minimal clinical benefit thresholds of total joint arthroplasty (TJA) (Hip Disability and Osteoarthritis Score Junior 30.5 vs 20.8, P&nbsp;= .23; Knee Injury and Osteoarthritis Outcome Score Junior 23.3 vs 18.2, P&nbsp;= .77) within 6 weeks. Patients who objectively "take it easy" after TJA may experience less pain with no difference in early subjective outcome. Remote, passive analysis of outpatient wearable sensor data may permit real-time detection of early problems after TJA

A computational framework for two-dimensional random walks with restarts

The treatment of two-dimensional random walks in the quarter plane leads to Markov processes which involve semi-infinite matrices having Toeplitz or block Toeplitz structure plus a low-rank correction. Finding the steady state probability distribution of the process requires to perform operations involving these structured matrices. We propose an extension of the framework of [5] which allows to deal with more general situations such as processes involving restart events. This is motivated by the need for modeling processes that can incur in unexpected failures like computer system reboots. Algebraically, this gives rise to corrections with infinite support that cannot be treated using the tools currently available in the literature. We present a theoretical analysis of an enriched Banach algebra that, combined with appropriate algorithms, enables the numerical treatment of these problems. The results are applied to the solution of bidimensional Quasi-Birth-Death processes with infinitely many phases which model random walks in the quarter plane, relying on the matrix analytic approach. This methodology reduces the problem to solving a quadratic matrix equation with coefficients of infinite size. We provide conditions on the transition probabilities which ensure that the solution of interest of the matrix equation belongs to the enriched algebra. The reliability of our approach is confirmed by extensive numerical experimentation on some case studies

Implementing a perioperative efficiency initiative for orthopedic surgery instrumentation at an academic center: A comparative before-and-after study.

Optimizing surgical instrumentation may contribute to value-based care, particularly in commonly performed procedures. We report our experience in implementing a perioperative efficiency program in 2 types of orthopedic surgery (primary total-knee arthroplasty, TKA, and total-hip arthroplasty, THA).A comparative before-and-after study with 2 participating surgeons, each performing both THA and TKA, was conducted. Our objective was to evaluate the effect of surgical tray optimization on operating and processing time, cost, and waste associated with preparation, delivery, and staging of sterile surgical instruments. The study was designed as a prospective quality improvement initiative with pre- and postimplementation operational measures and a provider satisfaction survey.A total of 96 procedures (38 preimplementation and 58 postimplementation) were assessed using time-stamped performance endpoints. The number and weight of trays and instruments processed were reduced substantially after the optimization intervention, particularly for TKA. Setup time was reduced by 23% (6 minutes, P = .01) for TKA procedures but did not differ for THA. The number of survey respondents was small, but satisfaction was high overall among personnel involved in implementation.Optimizing instrumentation trays for orthopedic procedures yielded reduction in processing time and cost. Future research should evaluate patient outcomes and incremental/additive impact on institutional quality measures

Digital Orthopaedics: A Glimpse Into the Future in the Midst of a Pandemic.

BackgroundThe response to COVID-19 catalyzed the adoption and integration of digital health tools into the health care delivery model for musculoskeletal patients. The change, suspension, or relaxation of Medicare and federal guidelines enabled the rapid implementation of these technologies. The expansion of payment models for virtual care facilitated its rapid adoption. The authors aim to provide several examples of digital health solutions utilized to manage orthopedic patients during the pandemic and discuss what features of these technologies are likely to continue to provide value to patients and clinicians following its resolution.ConclusionThe widespread adoption of new technologies enabling providers to care for patients remotely has the potential to permanently change the expectations of all stakeholders about the way care is provided in orthopedics. The new era of Digital Orthopaedics will see a gradual and nondisruptive integration of technologies that support the patient's journey through the successful management of their musculoskeletal disease