Weakly interlacing polynomials and totally nonnegative Hurwitz matrices

Well-known theorems in classical analysis tell us that a pair $p_0$, $p_1$ of real polynomials with positive coefficients have simple, interlacing roots if and only if all nontrivial minors of a certain infinite matrix associated to $p_0$ and $p_1$ -- called the Hurwitz matrix $H(p_0,p_1)$ -- are strictly positive. What happens if we relax one side of this equivalence, allowing $p_0$ and $p_1$ to weakly interlace, i.e. have shared and/or repeated roots? It has been shown in this case that $H(p_0,p_1)$ is still totally nonnegative. In the present work, we prove the converse statement: if $H(p_0,p_1)$ is totally nonnegative, then $p_0$ and $p_1$ weakly interlace. Thus our result completes the analogy between the strict and weak versions of this equivalence. The proof employs Temperley-Lieb immanants, first introduced by Rhoades and Skandera.Comment: 21 page

Chow Rings of Matroids as Permutation Representations

Given a matroid and a group of its matroid automorphisms, we study the induced group action on the Chow ring of the matroid. This turns out to always be a permutation action. Work of Adiprasito, Huh and Katz showed that the Chow ring satisfies Poincar\'e duality and the Hard Lefschetz theorem. We lift these to statements about this permutation action, and suggest further conjectures in this vein.Comment: 21 pages, 3 figure

Use of Novel Strategies to Develop Guidelines for Management of Pyogenic Osteomyelitis in Adults: A WikiGuidelines Group Consensus Statement.

Importance Traditional approaches to practice guidelines frequently result in dissociation between strength of recommendation and quality of evidence. Objective To construct a clinical guideline for pyogenic osteomyelitis management, with a new standard of evidence to resolve the gap between strength of recommendation and quality of evidence, through the use of a novel open access approach utilizing social media tools. Evidence Review This consensus statement and systematic review study used a novel approach from the WikiGuidelines Group, an open access collaborative research project, to construct clinical guidelines for pyogenic osteomyelitis. In June 2021 and February 2022, authors recruited via social media conducted multiple PubMed literature searches, including all years and languages, regarding osteomyelitis management; criteria for article quality and inclusion were specified in the group's charter. The GRADE system for evaluating evidence was not used based on previously published concerns regarding the potential dissociation between strength of recommendation and quality of evidence. Instead, the charter required that clear recommendations be made only when reproducible, prospective, controlled studies provided hypothesis-confirming evidence. In the absence of such data, clinical reviews were drafted to discuss pros and cons of care choices. Both clear recommendations and clinical reviews were planned with the intention to be regularly updated as new data become available. Findings Sixty-three participants with diverse expertise from 8 countries developed the group's charter and its first guideline on pyogenic osteomyelitis. These participants included both nonacademic and academic physicians and pharmacists specializing in general internal medicine or hospital medicine, infectious diseases, orthopedic surgery, pharmacology, and medical microbiology. Of the 7 questions addressed in the guideline, 2 clear recommendations were offered for the use of oral antibiotic therapy and the duration of therapy. In addition, 5 clinical reviews were authored addressing diagnosis, approaches to osteomyelitis underlying a pressure ulcer, timing for the administration of empirical therapy, specific antimicrobial options (including empirical regimens, use of antimicrobials targeting resistant pathogens, the role of bone penetration, and the use of rifampin as adjunctive therapy), and the role of biomarkers and imaging to assess responses to therapy. Conclusions and Relevance The WikiGuidelines approach offers a novel methodology for clinical guideline development that precludes recommendations based on low-quality data or opinion. The primary limitation is the need for more rigorous clinical investigations, enabling additional clear recommendations for clinical questions currently unresolved by high-quality data

A retrospective analysis of cases of Spontaneous Bacterial Peritonitis in cirrhosis patients.

Background & aimsSpontaneous Bacterial Peritonitis (SBP) is an infection in patients with cirrhosis and carries significant mortality. The management of SBP is evolving with the rise of multidrug resistant organisms. Our aim was to perform a retrospective analysis to determine if identification of bacteria in culture could aid in prognosis and provide information regarding optimal treatment.MethodsWe analyzed our 10-year experience of SBP in a single academic center (Northwestern Memorial Hospital). We obtained information regarding SBP prophylaxis, culture data and resistance patterns of bacteria, choice/duration of inpatient antibiotics, and key laboratory measurements and determined outcomes including mortality, hospital duration, and ICU stay.ResultsPatients with SBP had a 17.8% mortality and had culture positive SBP 34.4% of the time. Antimicrobial resistance was seen in 21.3% of cases and trended towards worsening mortality, with worsened mortality associated with first line use of piperacillin-tazobactam (p = 0.0001). Patients on SBP prophylaxis who developed SBP had improved mortality (pConclusionsOur results show that SBP continues to be a morbid and deadly condition and identification of an organism is key in treatment. The standard initial antibiotic for SBP may need to be modified to reflect emerging resistant pathogens and gram-positive organisms. Further, antibiotic prophylaxis should be utilized only in select cases to prevent development of resistance