On simultaneous diagonalization via congruence of real symmetric matrices

Simultaneous diagonalization via congruence (SDC) for more than two symmetric matrices has been a long standing problem. So far, the best attempt either relies on the existence of a semidefinite matrix pencil or casts on the complex field. The problem now is resolved without any assumption. We first propose necessary and sufficient conditions for SDC in case that at least one of the matrices is nonsingular. Otherwise, we show that the singular matrices can be decomposed into diagonal blocks such that the SDC of given matrices becomes equivalently the SDC of the sub-matrices. Most importantly, the sub-matrices now contain at least one nonsingular matrix. Applications to simplify some difficult optimization problems with the presence of SDC are mentioned

Commune Health Workers' Methadone Maintenance Treatment (MMT) Knowledge and Perceived Difficulties Providing Decentralized MMT Services in Vietnam.

BackgroundWith the initial establishment of countrywide methadone maintenance therapy (MMT) system, Vietnam is in the process of expanding and decentralizing the MMT program to community-based healthcare settings.ObjectiveThe study aimed to measure the MMT-related knowledge and perceived difficulties in treating patient who use drugs (PWUD) among community-based healthcare providers, e.g., commune health workers (CHW), and examine its correlated factors.MethodsA total of 300 CHW from 60 communes in two provinces of Vietnam completed a survey using Audio Computer-Assisted Self-Interview (ACASI) method. Twelve true-or-false questions were used to assess the CHW's MMT-related knowledge. The CHW's background characteristics and perceived difficulties treating PWUD were recorded.ResultsThe mean MMT knowledge score was 8.2 (SD = 1.2; range: 5-11). Misconceptions toward the benefits, procedure, and side effects of MMT were prevalent. The participants perceived varying degrees of difficulties in recruiting, engaging, and communicating with PWUD. With all covariates holding constant, younger age (standardized ẞ = -0.166; p = 0.0078) was associated with less MMT-related knowledge. Number of PWUD seen in a month and MMT-related knowledge was associated with less perceived difficulties treating PWUD. Conclusions/importance: The finding shed lights on the CHW's knowledge gap, which need to be addressed to facilitate the decentralization of MMT services in Vietnam. In preparation for a decentralized MMT service delivery model, specially designed training is warranted to equip CHW with knowledge and confidence to provide MMT-related services to PWUD

Correlated outcomes of a pilot intervention for people injecting drugs and their family members in Vietnam.

BackgroundThe interrelationship between the well-being of injecting drug users (IDUs) and their family environment has been widely documented. However, few intervention programs have addressed the needs of both IDUs and their family members.MethodsThis study describes a randomized intervention pilot targeting 83 IDUs and 83 of their family members from four communes in Phú Thọ province, Vietnam. The IDUs and family members in the intervention condition received multiple group sessions, with the intent to improve psychological well-being and family relationships. The intervention outcomes (depressive symptoms and family relations) were evaluated at baseline, 3-month and 6-month follow-up assessments.ResultsDepressive symptoms and family relations reported by IDUs were found to be correlated to those reported by their family members. Overall, significant intervention effects on depressive symptoms and family relations were observed for both IDUs and family members. A similar improvement pattern in family relations emerged for both the IDU and family member samples, although the intervention effect of reducing depressive symptoms was more sustainable for family members at the 6-month assessment when compared to the IDU sample.ConclusionThe intervention pilot addressed challenges faced by IDUs and their family members and revealed correlated outcomes for the two groups. Findings suggest a vital need to include family members in future drug prevention and harm reduction intervention efforts

An SDP Approach For Solving Quadratic Fractional Programming Problems

This paper considers a fractional programming problem (P) which minimizes a ratio of quadratic functions subject to a two-sided quadratic constraint. As is well-known, the fractional objective function can be replaced by a parametric family of quadratic functions, which makes (P) highly related to, but more difficult than a single quadratic programming problem subject to a similar constraint set. The task is to find the optimal parameter $\lambda^*$ and then look for the optimal solution if $\lambda^*$ is attained. Contrasted with the classical Dinkelbach method that iterates over the parameter, we propose a suitable constraint qualification under which a new version of the S-lemma with an equality can be proved so as to compute $\lambda^*$ directly via an exact SDP relaxation. When the constraint set of (P) is degenerated to become an one-sided inequality, the same SDP approach can be applied to solve (P) {\it without any condition}. We observe that the difference between a two-sided problem and an one-sided problem lies in the fact that the S-lemma with an equality does not have a natural Slater point to hold, which makes the former essentially more difficult than the latter. This work does not, either, assume the existence of a positive-definite linear combination of the quadratic terms (also known as the dual Slater condition, or a positive-definite matrix pencil), our result thus provides a novel extension to the so-called "hard case" of the generalized trust region subproblem subject to the upper and the lower level set of a quadratic function.Comment: 26 page

Enzymatic Cross-Linking of Dynamic Thiol-Norbornene Click Hydrogels

Enzyme-mediated in situ forming hydrogels are attractive for many biomedical applications because gelation afforded by enzymatic reactions can be readily controlled not only by tuning macromer compositions, but also by adjusting enzyme kinetics. For example, horseradish peroxidase (HRP) has been used extensively for in situ cross-linking of macromers containing hydroxyl-phenol groups. The use of HRP to initiate thiol-allylether polymerization has also been reported, yet no prior study has demonstrated enzymatic initiation of thiol-norbornene gelation. In this study, we discovered that HRP can generate the thiyl radicals needed for initiating thiol-norbornene hydrogelation, which has only been demonstrated previously using photopolymerization. Enzymatic thiol-norbornene gelation not only overcomes light attenuation issue commonly observed in photopolymerized hydrogels, but also preserves modularity of the cross-linking. In particular, we prepared modular hydrogels from two sets of norbornene-modified macromers, 8-arm poly(ethylene glycol)-norbornene (PEG8NB) and gelatin-norbornene (GelNB). Bis-cysteine-containing peptides or PEG-tetra-thiol (PEG4SH) was used as a cross-linker for forming enzymatically and orthogonally polymerized hydrogel. For HRP-initiated PEG-peptide hydrogel cross-linking, gelation efficiency was significantly improved via adding tyrosine residues on the peptide cross-linkers. Interestingly, these additional tyrosine residues did not form permanent dityrosine cross-links following HRP-induced gelation. As a result, they remained available for tyrosinase-mediated secondary cross-linking, which dynamically increased hydrogel stiffness. In addition to material characterizations, we also found that both PEG- and gelatin-based hydrogels exhibited excellent cytocompatibility for dynamic 3D cell culture. The enzymatic thiol-norbornene gelation scheme presented here offers a new cross-linking mechanism for preparing modularly and dynamically cross-linked hydrogels