A Grassmann integral equation

The present study introduces and investigates a new type of equation which is called Grassmann integral equation in analogy to integral equations studied in real analysis. A Grassmann integral equation is an equation which involves Grassmann integrations and which is to be obeyed by an unknown function over a (finite-dimensional) Grassmann algebra G_m. A particular type of Grassmann integral equations is explicitly studied for certain low-dimensional Grassmann algebras. The choice of the equation under investigation is motivated by the effective action formalism of (lattice) quantum field theory. In a very general setting, for the Grassmann algebras G_2n, n = 2,3,4, the finite-dimensional analogues of the generating functionals of the Green functions are worked out explicitly by solving a coupled system of nonlinear matrix equations. Finally, by imposing the condition G[{\bar\Psi},{\Psi}] = G_0[{\lambda\bar\Psi}, {\lambda\Psi}] + const., 0<\lambda\in R (\bar\Psi_k, \Psi_k, k=1,...,n, are the generators of the Grassmann algebra G_2n), between the finite-dimensional analogues G_0 and G of the (``classical'') action and effective action functionals, respectively, a special Grassmann integral equation is being established and solved which also is equivalent to a coupled system of nonlinear matrix equations. If \lambda \not= 1, solutions to this Grassmann integral equation exist for n=2 (and consequently, also for any even value of n, specifically, for n=4) but not for n=3. If \lambda=1, the considered Grassmann integral equation has always a solution which corresponds to a Gaussian integral, but remarkably in the case n=4 a further solution is found which corresponds to a non-Gaussian integral. The investigation sheds light on the structures to be met for Grassmann algebras G_2n with arbitrarily chosen n.Comment: 58 pages LaTeX (v2: mainly, minor updates and corrections to the reference section; v3: references [4], [17]-[21], [39], [46], [49]-[54], [61], [64], [139] added

GeoGebra Intervention: How have Students’ Performance and Confidence in Algebra Advanced?

The study’s goal was to provide an educational intervention in Algebra through GeoGebra that would boost students’ confidence, improve their learning, and correct their most minor mastered skills, allowing them to improve their Algebra performance. The research design was quasi-experimental, with 40 nonrandomly chosen participants comprising the GeoGebra and control groups. Mean and standard deviation was employed to describe the algebra performance and confidence of the respondents. At the same time, independent and dependent t-tests were used to determine the students’ significant difference in algebra performance and confidence in the pre-and post-test between the control and GeoGebra groups. GeoGebra effectively improved algebra confidence, enhanced learning, and remedied the students’ least mastered skills. GeoGebra is recommended as an instructional material in teaching and learning Algebra and can be extended to other mathematics content areas

Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations

We review the application of mathematical modeling to understanding the behavior of populations of chemotactic bacteria. The application of continuum mathematical models, in particular generalized Keller–Segel models, is discussed along with attempts to incorporate the microscale (individual) behavior on the macroscale, modeling the interaction between different species of bacteria, the interaction of bacteria with their environment, and methods used to obtain experimentally verified parameter values. We allude briefly to the role of modeling pattern formation in understanding collective behavior within bacterial populations. Various aspects of each model are discussed and areas for possible future research are postulated

Cardiovascular and metabolic health is associated with functional brain connectivity in middle-aged and older adults: Results from the Human Connectome Project-Aging study

Several cardiovascular and metabolic indicators, such as cholesterol and blood pressure have been associated with altered neural and cognitive health as well as increased risk of dementia and Alzheimer\u27s disease in later life. In this cross-sectional study, we examined how an aggregate index of cardiovascular and metabolic risk factor measures was associated with correlation-based estimates of resting-state functional connectivity (FC) across a broad adult age-span (36-90+ years) from 930 volunteers in the Human Connectome Project Aging (HCP-A). Increased (i.e., worse) aggregate cardiometabolic scores were associated with reduced FC globally, with especially strong effects in insular, medial frontal, medial parietal, and superior temporal regions. Additionally, at the network-level, FC between core brain networks, such as default-mode and cingulo-opercular, as well as dorsal attention networks, showed strong effects of cardiometabolic risk. These findings highlight the lifespan impact of cardiovascular and metabolic health on whole-brain functional integrity and how these conditions may disrupt higher-order network integrity

Globalization, Roundaboutness, and Relative Wages

We depart from the trade and wages literature and its emphasis on North-South trade, examining North-North by developing the basic linkages between trade-based integration and relative wages in an Ethier-type division of labor model. Using this model we identify a formal relationship between international trade, productivity, and wages. We then examine the trivariate relationship between trade, growth in total factor productivity (TFP), and the skill premium in a vector autoregression framework. We find evidence of a long-run relationship between growth in intermediate goods and changes in TFP. Controlling for this relationship we also find a positive relationship between trade and the skill-premium

Haloarchaea swim slowly for optimal chemotactic efficiency in low nutrient environments

Archaea have evolved to survive in some of the most extreme environments on earth. Life in extreme, nutrient-poor conditions gives the opportunity to probe fundamental energy limitations on movement and response to stimuli, two essential markers of living systems. Here we use three-dimensional holographic microscopy and computer simulations to reveal that halophilic archaea achieve chemotaxis with power requirements one hundred-fold lower than common eubacterial model systems. Their swimming direction is stabilised by their flagella (archaella), enhancing directional persistence in a manner similar to that displayed by eubacteria, albeit with a different motility apparatus. Our experiments and simulations reveal that the cells are capable of slow but deterministic chemotaxis up a chemical gradient, in a biased random walk at the thermodynamic limit

ECCO Essential Requirements for Quality Cancer Care: Soft Tissue Sarcoma in Adults and Bone Sarcoma. A critical review

ECCO essential requirements for quality cancer care (ERQCC) are checklists and explanations of organisation and actions that are necessary to give high-quality care to patients who have a specific tumour type. They are written by European experts representing all disciplines involved in cancer care. ERQCC papers give oncology teams, patients, policymakers and managers an overview of the elements needed in any healthcare system to provide high quality of care throughout the patient journey. References are made to clinical guidelines and other resources where appropriate, and the focus is on care in Europe. Sarcoma: essential requirements for quality care • Sarcomas – which can be classified into soft tissue and bone sarcomas – are rare, but all rare cancers make up more than 20% of cancers in Europe, and there are substantial inequalities in access to high-quality care. Sarcomas, of which there are many subtypes, comprise a particularly complex and demanding challenge for healthcare systems and providers. This paper presents essential requirements for quality cancer care of soft tissue sarcomas in adults and bone sarcomas. • High-quality care must only be carried out in specialised sarcoma centres (including paediatric cancer centres) which have both a core multidisciplinary team and an extended team of allied professionals, and which are subject to quality and audit procedures. Access to such units is far from universal in all European countries. • It is essential that, to meet European aspirations for high-quality comprehensive cancer control, healthcare organisations implement the requirements in this paper, paying particular attention to multidisciplinarity and patient-centred pathways from diagnosis and follow-up, to treatment, to improve survival and quality of life for patients. Conclusion Taken together, the information presented in this paper provides a comprehensive description of the essential requirements for establishing a high-quality service for soft tissue sarcomas in adults and bone sarcomas. The ECCO expert group is aware that it is not possible to propose a ‘one size fits all’ system for all countries, but urges that access to multidisciplinary teams is guaranteed to all patients with sarcoma