Classification of Multidimensional Darboux Transformations: First Order and Continued Type

We analyze Darboux transformations in very general settings for multidimensional linear partial differential operators. We consider all known types of Darboux transformations, and present a new type. We obtain a full classification of all operators that admit Wronskian type Darboux transformations of first order and a complete description of all possible first-order Darboux transformations. We introduce a large class of invertible Darboux transformations of higher order, which we call Darboux transformations of continued Type I. This generalizes the class of Darboux transformations of Type I, which was previously introduced. There is also a modification of this type of Darboux transformations, continued Wronskian type, which generalize Wronskian type Darboux transformations

Generalized pressure drop and heat transfer correlations for jet impingement cooling with jet adjacent fluid extraction

2022 Spring.Includes bibliographical references.Jet impingement technologies offer a promising solution to thermal management challenges across multiple fields and applications. Single jets and conventional impinging arrays have been studied extensively and are broadly recognized for achieving extraordinary local heat transfer coefficients. This, in combination with the versatility of impinging arrays, has facilitated a steady incline in the popularity of jet impingement investigations. However, it is well documented that interactions between adjacent jets in an impinging array have a debilitating effect on thermal performance. Recently, in an attempt to mitigate the jet interference problem, a number of researchers have created innovative jet impingement solutions which eliminate crossflow effects by introducing fluid extraction ports interspersed throughout the impinging array. This novel adaptation on classical impinging arrays has been shown to produce dramatically improved thermal performance and offers an excellent opportunity for future high-performing thermal management devices. The advent of jet-adjacent fluid extraction in impinging arrays presents a promising improvement to impingement cooling technologies. However, there have been very few investigations to quantify these effects. Notably, the current archive of literature is severely lacking in useful, predictive correlations for heat transfer and pressure drop which can reliably describe the performance of such impinging arrays. Steady-state heat transfer and adiabatic pressure drop experiments were conducted using nine unique geometric configurations of a novel jet impingement device developed in this work. This investigation proposes novel empirical correlations for Darcy friction factor and Nusselt number in an impingement array with interspersed fluid extraction ports. The correlations cover a broad range of geometric parameters, including non-dimensional jet array spacing (S/Dj) ranging from 2.7 to 9.1, and non-dimensional jet heights (H/Dj) ranging from 0.31 to 4.4. Experiments included jet Reynolds numbers ranging from 70 to 24,000, incorporating laminar and turbulent flow regimes. Multiple fluids were tested with Prandtl numbers ranging from 0.7 to 21. The correlations presented in this work are the most comprehensive to date for impinging jet arrays with interspersed fluid extraction. Nusselt number was found to be correlated to impinging jet Reynolds number to the power of 0.57. The resulting correlation was able to predict 93% of experimental data within ±25%. During adiabatic pressure drop experiments, multiple laminar-turbulent flow transition regions were identified at various stages in the complex jet impingement flow path. The proposed Darcy friction factor correlation was separated into laminar, turbulent, and transition regions and predicted experimental data with a mean absolute deviation of 20%. The heat transfer and pressure drop correlations proposed in this investigation were used in a follow-on optimization study which targeted an exemplary impingement cooling application. The optimization study applied core experimental findings to a microchip cooling case study and evaluated the effects of geometry, flow, and heat load parameters on cooling efficiency and effectiveness. It was discovered that reducing non-dimensional jet height results in all-around improved cooling performance. Conversely, low non-dimensional jet spacing results in highly efficient but less effective solutions while high non-dimensional jet spacing results in effective but less efficient cooling

Nontrivial solutions for homogeneous linear equations over some non-quotient hyperfields

We introduce a class of hyperfields which includes several constructions of non-quotient hyperfields. We then use it to partially answer a question posed by M. Baker and T. Zhang: Does a system of homogeneous linear equations with more unknowns than equations always have a nonzero solution? We also consider a class of hyperfields that was claimed in the literature to be non-quotient, and show that this is false

Satisfiability in multi-valued circuits

Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras $\mathbf A$ there is a polynomial time algorithm that decides if an equation over $\mathbf A$ has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time.Comment: 50 page

Community organising frameworks, models, and processes to improve health : A systematic scoping review

Community involvement engages, empowers, and mobilises people to achieve their shared goals by addressing structural inequalities in the social and built environment. Through this review, we summarised published information on models, frameworks, and/or processes of community organising used in the context of health initiatives or interventions and documented the outcomes following their use. A systematic scoping review was conducted in three databases with no restrictions on the date of publication, country, or written language. Out of 5044 studies, 38 met the inclusion criteria and were included in the review. The targeted health outcomes explored by the studies were diverse and included sub-domains such as the promotion of a healthy lifestyle, sexual and reproductive health, access to healthcare and equity, and substance abuse and chronic disease management. The outcomes of most initiatives or interventions were promising, with positive changes reported for the target populations. A wide variation was noted in the models, frameworks, or processes of community organising utilised in these studies. We concluded that variation implies that no single model, framework, or process seems to have predominance over others in implementing community organising as a vehicle of positive social change within the health domain. The review also highlighted the need for a more standardised approach to the implementation and evaluation of these initiatives. We recommend that it is essential to foster public and non-governmental sector partnerships to promote community-driven health promotion efforts for a more sustainable approach to these initiatives

Referrals from community optometrists to the hospital eye service in Scotland and England

Objectives: This audit assesses communication between community optometrists (COs) and hospital eye service (HES) in Scotland and England. Methods: Optometric referrals and replies were extracted from six practices in Scotland and England. If no reply was found, replies/records were copied from HES records. De-identified referrals, replies and records were audited against established standards, evaluating whether referrals were necessary, accurate and directed to the appropriate professional. The referral rate (RR) and referral reply rate (RRR) were calculated. Results: From 905 de-identified referrals, RR ranged from 2.6 to 8.7%. From COs’ perspective, the proportion of referrals for which they received replies ranged from 37 to 84% (Scotland) and 26 to 49% (England). A total of 88–96% of referrals (Scotland) and 63–76% (England) were seen in the HES. Adjusting for cases when it is reasonable to expect replies, RRR becomes 45–92% (Scotland) and 38–62% (England) with RRR significantly greater in Scotland (P = 0.015). Replies were copied to patients in 0–21% of cases. Referrals were to the appropriate service and judged necessary in ≥90% of cases in both jurisdictions. Accuracy of referral ranged from 89 to 97% (Scotland) and 81 to 98% (England). The reply addressed the reason for referral in 94–100% of cases (Scotland) and 93–97% (England) and was meaningful in 95–100% (Scotland) and 94–99% (England). Conclusions: Despite the interdisciplinary joint statement on sharing patient information, this audit highlights variable standard of referrals and deficits in replies to the referring COs, with one exception in Scotland. Replies from HES to COs are important for patient care, benefitting patients and clinicians and minimising unnecessary HES appointments

Detection of Mycobacterium tuberculosis in Sputum by Gas Chromatography-Mass Spectrometry of Methyl Mycocerosates Released by Thermochemolysis

Tuberculosis requires rapid diagnosis to prevent further transmission and allow prompt administration of treatment. Current methods for diagnosing pulmonary tuberculosis lack sensitivity are expensive or are extremely slow. The identification of lipids using gas chromatography- electron impact mass spectrometry (GC-EI/MS) could provide an alternative solution. We have studied mycocerosic acid components of the phthiocerol dimycocerosate (PDIM) family of lipids using thermochemolysis GC-EI/MS. To facilitate use of the technology in a routine diagnostic laboratory a simple extraction procedure was employed where PDIMs were extracted from sputum using petroleum ether, a solvent of low polarity. We also investigated a method using methanolic tetramethylammonium hydroxide, which facilitates direct transesterification of acidic components to methyl esters in the inlet of the GC-MS system. This eliminates conventional chemical manipulations allowing rapid and convenient analysis of samples. When applied to an initial set of 40 sputum samples, interpretable results were obtained for 35 samples with a sensitivity relative to culture of 94% (95%CI: 69.2,100) and a specificity of 100% (95%CI: 78.1,100). However, blinded testing of a larger set of 395 sputum samples found the assay to have a sensitivity of 61.3% (95%CI: 54.9,67.3) and a specificity of 70.6% (95%CI: 62.3,77.8) when compared to culture. Using the results obtained we developed an improved set of classification criteria, which when applied in a blinded re-analysis increased the sensitivity and specificity of the assay to 64.9% (95%CI: 58.6,70.8) and 76.2% (95%CI: 68.2,82.8) respectively. Highly variable levels of background signal were observed from individual sputum samples that inhibited interpretation of the data. The diagnostic potential of using thermochemolytic GC-EI/MS of PDIM biomarkers for diagnosis of tuberculosis in sputum has been established; however, further refinements in sample processing are required to enhance the sensitivity and robustness of the test