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Mathematician and popular science author Eugenia Cheng is on a mission to show you that mathematics can be flexible, creative, and visual. This joyful journey through the world of abstract mathematics into category theory will demystify mathematical thought processes and help you develop your own thinking, with no formal mathematical background needed. The book brings abstract mathematical ideas down to earth using examples of social justice, current events, and everyday life – from privilege to COVID-19 to driving routes. The journey begins with the ideas and workings of abstract mathematics, after which you will gently climb toward more technical material, learning everything needed to understand category theory, and then key concepts in category theory like natural transformations, duality, and even a glimpse of ongoing research in higher-dimensional category theory. For fans of How to Bake Pi, this will help you dig deeper into mathematical concepts and build your mathematical background.
A partially-auxetic metamaterial is introduced, inspired by the Maltese cross. Each unit of this metamaterial consists of a pair of counter-rotating equal-armed crosses, which is interconnected to neighboring units via hinge rods and connecting rods. Based on linkage theory, the on-axes Poisson's ratio was established considering a two-fold symmetrical mechanism, while the (anti)tetrachiral mechanisms were identified for on-axes uniaxial compression. A shearing mechanism is suggested for pure shearing and diagonal loading of the metamaterial with square array. Results suggest that the approximated infinitesimal models are valid for the Poisson's ratio of the two-fold symmetrical and the (anti)tetrachiral mechanisms under on-axis tension and compression, respectively; however, the finite model is recommended for quantifying the Poisson's ratio under pure shear and off-axis loading. This metamaterial manifests microstructural trinity, in which three different loading modes result in three different groups of deformation mechanisms. Finally, suggestions are put forth for some unsolved predictive problems.
Difference and disagreement can be valuable, yet they can also spiral out of control and damage liberal democracy. Advancing a metaphor of citizenship that the author terms 'role-based constitutional fellowship,' this book offers a solution to this challenge. Cheng argues that a series of 'divisions of labor' among citizens, differently situated, can help cultivate the foundational trust required to harness the benefits of disagreement and difference while preventing them from 'overheating' and, in turn, from leaving liberal democracy vulnerable to the growing influence of autocratic political forces. The book recognizes, however, that it is not always appropriate to attempt to cultivate trust, and acknowledges the important role that some forms of confrontation might play in identifying and rectifying undue social hierarchies, such as racial-ethnic hierarchies. Hanging Together thereby works to pave a middle way between deliberative and realist conceptions of democracy.
Issues of joint dependence between the dependent variables and explanatory variables are discussed. Error components 2SLS or 3SLS estimator for an equation or a system is introduced. Identification conditions with prior restrictions on the parameters or the disturbance terms are illustrated with a triangular system.
Issues of some widely used nonlinear models such as the duration, count data, or nonparametric estimation of general nonlinear models in the presence of individual or time-specific effects are discussed. Consistent and asymptotically normally distributed estimators are introduced.
Dynamic linear error-components models are introduced. Issues of multi-dimensional asymptotics are discussed. Large sample properties of the instrumental variable estimator (IV), generalized method of moments estimator (GMM), and maximum likelihood estimator (MLE) when N is fixed and T goes to infinity, or T is fixed and N goes to infinity, or both N and T large, are examined. Bias correction estimators are introduced.
Advantages and challenges of using factor structure to control the impact of unobserved heterogeneity that varies across individuals and over time are discussed. Methods for the determination of the dimension of factor structure are also discussed.
Weak and strong cross-sectional dependence are discussed. Various spatial approaches to model cross-sectional dependence, such as the spatial error or spatial regressive approach for static or dynamic models, are introduced. Issues of endogeneity of spatial weight matrix, higher-order spatial weight matrix, and the mixed spatial and factor process are considered. The Lagrangian multiplier and CD tests for cross-sectional uncorrelation for various types of models are also discussed.
Issues of nonrandom sampling due to truncation, censoring, or sample selection rules are discussed in the presence of individual-specific effects. Symmetric trimming of sample observations to get rid of incidental parameters are introduced.
The use of a panel vector autoregressive model as a reduced form approximation to a panel dynamic system is introduced. Nonstationarity and cointegrations over time and across cross-sections are discussed. Identification conditions and MLE or GMM estimation of dynamic simultaneous equation models are considered.
Issues of quantile regression, simulation methods, multi-level panel data, errors of measurement, distributed lag models when T is short, rotating or randomly missing data, repeated cross-sectional data, and discretizing unobserved heterogeneity are discussed.
The motivation for generalizing unobserved heterogeneity of varying parameter models is discussed. Various fixed or random varying parameters across cross-sectional units and over time models together with their respective inference procedures are introduced from both the sampling approach and the Bayesian approach. Issues of correlations between parameter variation and regressors are also discussed.