Constructions of biangular tight frames and their relationships with equiangular tight frames

We study several interesting examples of Biangular Tight Frames (BTFs) - basis-like sets of unit vectors admitting exactly two distinct frame angles (ie, pairwise absolute inner products) - and examine their relationships with Equiangular Tight Frames (ETFs) - basis-like systems which admit exactly one frame angle. We demonstrate a smooth parametrization BTFs, where the corresponding frame angles transform smoothly with the parameter, which "passes through" an ETF answers two questions regarding the rigidity of BTFs. We also develop a general framework of so-called harmonic BTFs and Steiner BTFs - which includes the equiangular cases, surprisingly, the development of this framework leads to a connection with the famous open problem(s) regarding the existence of Mersenne and Fermat primes. Finally, we construct a (chordally) biangular tight set of subspaces (ie, a tight fusion frame) which "Pl\"ucker embeds" into an ETF.Comment: 19 page

Special Supplement: Ethical and Policy Issues in Rehabilitation Medicine

The field of medical rehabilitation is relatively new, a product in great part of the rapid developments in medical science during and after the Second World War. Until recently, the ethical problems of this new field were neglected. There seemed to be more pressing concerns as rehabilitation medicine struggled to establish itself, somtimes in the face of considerable skepticism or hostility. There also seemed no pressing moral questions of the kind and intensity to be encountered, say, in high technology acute care medicine or genetic engineering. With eyes focused on the dramatic and wrenching problems, those in biomedical ethics could and did easily overlook the quiet, less obtrusive, issues of rehabilitation

Phase retrieval by hyperplanes

We show that a scalable frame does phase retrieval if and only if the hyperplanes of its orthogonal complements do phase retrieval. We then show this result fails in general by giving an example of a frame for $\mathbb R^3$ which does phase retrieval but its induced hyperplanes fail phase retrieval. Moreover, we show that such frames always exist in $\mathbb R^d$ for any dimension $d$. We also give an example of a frame in $\mathbb R^3$ which fails phase retrieval but its perps do phase retrieval. We will also see that a family of hyperplanes doing phase retrieval in $\mathbb R^d$ must contain at least $2d-2$ hyperplanes. Finally, we provide an example of six hyperplanes in $\mathbb R^4$ which do phase retrieval

Feasibility and Potential Benefits of a Self-Monitoring Enhanced Lifestyle Intervention to Prevent Excessive Gestational Weight Gain in Women Who Are Overweight or Obese

Objective To evaluate the feasibility and potential benefits of a self-monitoring enhanced lifestyle intervention to prevent excessive gestational weight gain in women who are overweight and obese. Design A one-group, prospective design involving 8 weeks of healthy eating and physical activity and self-monitoring of weight, nutrition, and walking. Setting Recruitment and enrollment in prenatal clinics and self-monitoring at home. Participants Women (N = 22) at 14 to 24 gestational weeks, with body mass indexes of 25 to 40 kg/m2, without medical and psychiatric diseases that affected cognition or walking. Methods Participants self-monitored weight and nutrition intake for the first 4 weeks and weight, nutrition intake, and walking in the second 4 weeks. Feasibility data were collected weekly (attrition, self-monitoring adherence, program safety, participant feedback) or at the end of Week 8 (satisfaction ratings). Potential benefits included weight, nutrition, and physical activity, measured at baseline (T1), the end of Week 4 (T2), or the end of Week 8 (T3). Results Attrition rates were 27.3% by T2 and 40.9% by T3. Adherence to log return was 100%. No adverse effects were noted, but food craving was persistent, and stress levels were high. Program satisfaction was high. Trends for improved activity and reduced trans fat consumption were seen. Conclusion Our findings indicate that the intervention is worthy of further development and testing with a randomized controlled trial

Classifying Coloring Graphs

Given a graph G, its k-coloring graph is the graph whose vertex set is the proper k-colorings of the vertices of G with two k-colorings adjacent if they differ at exactly one vertex. In this paper, we consider the question: Which graphs can be coloring graphs? In other words, given a graph H, do there exist G and k such that H is the k-coloring graph of G? We will answer this question for several classes of graphs and discuss important obstructions to being a coloring graph involving order, girth, and induced subgraphs

Medicinal Plant Extracts and Natural Compounds for the Treatment of Cutaneous Lupus Erythematosus: A Systematic Review

Cutaneous lupus erythematosus (CLE) is a group of autoimmune connective tissue disorders that significantly impact quality of life. Current treatment approaches typically use antimalarial medications, though patients may become recalcitrant. Other treatment options include general immunosuppressants, highlighting the need for more and more targeted treatment options. The purpose of this systematic review was to identify potential compounds that could be repurposed for CLE from natural products since many rheumatologic drugs are derived from natural products, including antimalarials. This study was registered with PROSPERO, the international prospective register of systematic reviews (registration number CRD42021251048). We comprehensively searched Ovid Medline, Cochrane Library, and Scopus databases from inception to April 27th, 2021. These terms included cutaneous lupus erythematosus; general plant, fungus, bacteria terminology; selected plants and plant-derived products; selected antimalarials; and JAK inhibitors. Our search yielded 13,970 studies, of which 1,362 were duplicates. We screened 12,608 abstracts, found 12,043 to be irrelevant, and assessed 565 full-text studies for eligibility. Of these, 506 were excluded, and 59 studies were included in the data extraction. The ROBINS-I risk of bias assessment tool was used to assess studies that met our inclusion criteria. According to our findings, several natural compounds do reduce inflammation in lupus and other autoimmune skin diseases in studies using in vitro methods, mouse models, and clinical observational studies, along with a few randomized clinical trials. Our study has cataloged evidence in support of potential natural compounds and plant extracts that could serve as novel sources of active ingredients for the treatment of CLE. It is imperative that further studies in mice and humans are conducted to validate these findings

Polyubiquitin binding to ABIN1 is required to prevent autoimmunity

The protein ABIN1 possesses a polyubiquitin-binding domain homologous to that present in nuclear factor kappa B (NF-kappa B) essential modulator (NEMO), a component of the inhibitor of NF-kappa B (I kappa B) kinase (IKK) complex. To address the physiological significance of polyubiquitin binding, we generated knockin mice expressing the ABIN1[D485N] mutant instead of the wild-type (WT) protein. These mice developed all the hallmarks of autoimmunity, including spontaneous formation of germinal centers, isotype switching, and production of autoreactive antibodies. Autoimmunity was suppressed by crossing to MyD88(-/-) mice, demonstrating that toll-like receptor (TLR)-MyD88 signaling pathways are needed for the phenotype to develop. The B cells and myeloid cells of the ABIN1[D485N] mice showed enhanced activation of the protein kinases TAK, IKK-alpha/beta, c-Jun N-terminal kinases, and p38 alpha mitogen-activated protein kinase and produced more IL-6 and IL-12 than WT. The mutant B cells also proliferated more rapidly in response to TLR ligands. Our results indicate that the interaction of ABIN1 with polyubiquitin is required to limit the activation of TLR-MyD88 pathways and prevent autoimmunity