Free polynomial Biholomorphisms between free spectrahedra

Linear Matrix Inequality (LMI) Domains are bounded free convex semialgebraic sets. Along with their commutative counterparts, they appear in a host of fields including convex optimization, semidefinite programming, convex real algebraic geometry and systems theory. We consider free polynomial mappings between LMI Domains, polynomial equivalence versus affine linear equivalence, and automorphisms for certain classes of domains. We produce a class of pairs of LMI domains that are polynomially, but not affine linearly, biholomorphic

Free bianalytic maps between spectrahedra and spectraballs in a generic setting

Given a tuple $E=(E_1,\dots,E_g)$ of $d\times d$ matrices, the collection of those tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that $\| \sum E_j\otimes X_j\|\le 1$ is called a spectraball $\mathcal B_E$. Likewise, given a tuple $B=(B_1,\dots,B_g)$ of $e\times e$ matrices the collection of tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that $I + \sum B_j\otimes X_j +\sum B_j^* \otimes X_j^*\succeq 0$ is a free spectrahedron $\mathcal D_B$. Assuming $E$ and $B$ are irreducible, plus an additional mild hypothesis, there is a free bianalytic map $p:\mathcal B_E\to \mathcal D_B$ normalized by $p(0)=0$ and $p'(0)=I$ if and only if $\mathcal B_E=\mathcal B_B$ and $B$ spans an algebra. Moreover $p$ is unique, rational and has an elegant algebraic representation.Comment: 19 page

Free potential functions

We show that the derivative of a noncommutative free analytic map must be free-curl free -- an analog of having zero curl. Moreover, under the assumption that the free domain is connected, this necessary condition is sufficient. Specifically, if $T$ is analytic free demilinear (linear in half its variables) map defined on a connected free domain then $DT(X,H)[K,0] = DT(X,K)[H,0]$ if and only if there exists an analytic free map $f$ such that $Df(X)[H] = T(X,H)$.Comment: 13 page

The effect of coronal splits on the structural stability of bi-condylar tibial plateau fractures: a biomechanical investigation

Introduction!#!Surgical treatment of bi-condylar tibial plateau fractures is still challenging due to the complexity of the fracture and the difficult surgical approach. Coronal fracture lines are associated with a high risk of fixation failure. However, previous biomechanical studies and fracture classifications have disregarded coronal fracture lines.!##!Materials and methods!#!This study aimed to develop a clinically relevant fracture model (Fracture C) and compare its mechanical behavior with the traditional Horwitz model (Fracture H). Twelve samples of fourth-generation tibia Sawbones were utilized to realize two fracture models with (Fracture C) or without (Fracture H) a coronal fracture line and both fixed with lateral locking plates. Loading of the tibial plateau was introduced through artificial femur condyles to cyclically load the fracture constructs until failure. Stiffness, fracture gap movements, failure loads as well as relative displacements and rotations of fracture fragments were measured.!##!Results!#!The presence of a coronal fracture line reduced fracture construct stiffness by 43% (p = 0.013) and decreased the failure load by 38% from 593 ± 159 to 368 ± 63 N (p = 0.016). Largest displacements were observed at the medial aspect between the tibial plateau and the tibial shaft in the longitudinal direction. Again, the presence of the coronal fracture line reduced the stability of the fragments and created increased joint incongruities.!##!Conclusions!#!Coronal articular fracture lines substantially affect the mechanical response of tibia implant structures specifically on the medial side. With this in mind, utilizing a clinically relevant fracture model for biomechanical evaluations regarding bi-condylar tibial plateau fractures is strongly recommended

Distal femur fractures: basic science and international perspectives

Distal femur fractures are challenging injuries to manage, and complication rates remain high. This article summarizes the international and basic science perspectives regarding distal femoral fractures that were presented at the 2022 Orthopaedic Trauma Association Annual Meeting. We review a number of critical concepts that can be considered to optimize the treatment of these difficult fractures. These include biomechanical considerations for distal femur fixation constructs, emerging treatments to prevent post-traumatic arthritis, both systemic and local biologic treatments to optimize nonunion management, the relative advantages and disadvantages of plate versus nail versus dual-implant constructs, and finally important factors which determine outcomes. A robust understanding of these principles can significantly improve success rates and minimize complications in the treatment of these challenging injuries