Viscosity Solutions: An Introduction to the basics of the theory

In this work, concepts that appear in the basic theory of viscosity solutions theory is surveyed. Structures of sub and super di.erentials and sub and super semijets, and concepts of generalized second derivative, generalized 'maximum principle' and generalized 'comparison principle' are studied. Basic properties of semiconvex functions and sup (Jensen's) convolutions are presented. Existence and uniqueness of solutions of the Dirichlet Problem for first and second order nonlinear elliptic partial differential equations is studied

Multiscale Feature Analysis of Salivary Gland Branching Morphogenesis

Pattern formation in developing tissues involves dynamic spatio-temporal changes in cellular organization and subsequent evolution of functional adult structures. Branching morphogenesis is a developmental mechanism by which patterns are generated in many developing organs, which is controlled by underlying molecular pathways. Understanding the relationship between molecular signaling, cellular behavior and resulting morphological change requires quantification and categorization of the cellular behavior. In this study, tissue-level and cellular changes in developing salivary gland in response to disruption of ROCK-mediated signaling by are modeled by building cell-graphs to compute mathematical features capturing structural properties at multiple scales. These features were used to generate multiscale cell-graph signatures of untreated and ROCK signaling disrupted salivary gland organ explants. From confocal images of mouse submandibular salivary gland organ explants in which epithelial and mesenchymal nuclei were marked, a multiscale feature set capturing global structural properties, local structural properties, spectral, and morphological properties of the tissues was derived. Six feature selection algorithms and multiway modeling of the data was performed to identify distinct subsets of cell graph features that can uniquely classify and differentiate between different cell populations. Multiscale cell-graph analysis was most effective in classification of the tissue state. Cellular and tissue organization, as defined by a multiscale subset of cell-graph features, are both quantitatively distinct in epithelial and mesenchymal cell types both in the presence and absence of ROCK inhibitors. Whereas tensor analysis demonstrate that epithelial tissue was affected the most by inhibition of ROCK signaling, significant multiscale changes in mesenchymal tissue organization were identified with this analysis that were not identified in previous biological studies. We here show how to define and calculate a multiscale feature set as an effective computational approach to identify and quantify changes at multiple biological scales and to distinguish between different states in developing tissues

An Empirical Test of the Concept of the Adaptively Intelligent Attitude

This study provides an empirical test of a previously proposed assertion that intelligence as adaptation has an attitudinal as well as an ability component. The ability component deals with what the basic knowledge and skills are that underlie intelligence, and how much of each one an individual has. The attitudinal component deals with how an individual chooses to deploy the abilities they have. In other words, to what use are the abilities put? It is argued that it is impossible fully to separate the measurement of the ability component from the attitudinal one. In a diverse population, even taking an intelligence test will show itself to involve an attitude toward the test, which may enhance or detract from performance, as when one sees the test as irrelevant or harmful to one’s life, or as a sociocultural misfit to one’s life experience. To succeed, people need not only to have abilities, but attitudes that put those abilities to effective use to accomplish individuals’ life goals. In the study, we found that intelligent attitudes are related, but non-identical, to germane constructs, such as wisdom, the need for cognition, creativity, and openness to experience. Scores on the attitudinal measure were not related to scores on tests of fluid intelligence and academic abilities/achievement. Thus, the range of attitudes regarding how to deploy intelligence can vary over ability levels

Bipartite graph analysis.

<p>The changes in the correlation clusters of the four tissue samples are studied through bi-partite graph analysis for the untreated vs. treated epithelial tissue comparison in (a) and for the untreated vs. treated mesenchymal tissue comparison in (b).</p

Generation of Cell Graphs.

<p>Stitched images were segmented using the active contour method to define epithelial (white) vs mesenchymal tissue (black) in control (a) and ROCK inhibitor-treated explants (d). These masks were used to identify the epithelial nuclei (b, e) and mesenchymal nuclei (c, f). Using each nucleus as a vertex, cell-graphs were constructed for control and ROCK inhibitor-treated tissues, respectively (g, h), where zoomed regions of cell graphs corresponding to regions of the original images (shown as red boxes in a and d) are shown in detail. Epithelial tissue is respresented by the blue graph and the mesenchymal tissue is represented by the red graph. We discarded the sublingual tissues and only used the submandibilar gland, (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0032906#pone.0032906.s002" target="_blank">Figure S2</a>).</p

Local structural features.

<p>Local structural features.</p

Spectral features.

<p>Spectral features.</p

Control vs ROCK-inhibitor-treated comparison of mesenchymal tissues.

<p>Control vs ROCK-inhibitor-treated comparison of mesenchymal tissues.</p

Morphological (shape based) features.

<p>Morphological (shape based) features.</p