ABSTRACT
We consider polynomials of bi-degree (n, 1) over the skew field of quaternions where the indeterminates commute with each other and with all coefficients. Polynomials of this type do not generally admit factorizations. We recall a necessary and sufficient condition for existence of a factorization with univariate linear factors that has originally been stated by Skopenkov and Krasauskas. Such a factorization is, in general, non-unique by known factorization results for univariate quaternionic polynomials. We unveil existence of bivariate polynomials with non-unique factorizations that cannot be explained in this way and characterize them geometrically and algebraically. Existence of factorizations is related to the existence of special rulings of two different types (left/right) on the ruled surface parameterized by the bivariate polynomial in the projective space over the quaternions. Special non-uniqueness in above sense can be explained algebraically by commutation properties of factors in suitable factorizations. A necessary geometric condition for this to happen is degeneration to a point of at least one of the left/right rulings.
ABSTRACT
The aim of this study was to compare the effectiveness of barbed versus smooth sutures for soft tissue closure of exposed jawbone sites in medication-related osteonecrosis of the jaw (MRONJ) patients. Exposed necrotic jawbone sites surgically managed by intraoral soft tissue closure were evaluated. Either barbed sutures (Stratafix™ or V-Loc™) together with Prolene® or Vicryl® sutures were used. We estimated the effect of barbed sutures (BS) with Prolene® compared to smooth sutures (Vicryl®) on the hazard rate of intraoral soft tissue dehiscence using a multivariate Cox regression model within a target trial framework, adjusting for relevant confounders. In total, 306 operations were performed in 188 sites. In the primary analysis 182 sites without prior surgery were included. Of these, 113 sites developed a dehiscence during follow-up. 84 sites were operated using BS and Prolene®. A total of 222 sites were operated with Vicryl® (control group). In the BS group, the median time to event (i.e., dehiscence) was 148 days (interquartile range (IQR), 42-449 days) compared to 15 days (IQR, 12-52 days) in the control group. The hazard rate of developing intraoral dehiscence was 0.03 times (95%-confidence interval (CI): 0.01; 0.14, p < 0.001) lower for BS patients compared to the control group. Within the limits of a retrospective study, BS showed a high success rate and are therefore recommended for soft tissue closure of exposed jawbone sites in MRONJ patients. Additional studies are warranted to further evaluate this novel application of BS.
