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"Plane Geometry"
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All‐Solid‐State Planar Sodium‐Ion Microcapacitors with Multidirectional Fast Ion Diffusion Pathways
With the relentless development of smart and miniaturized electronics, the worldwide thirst for microscale electrochemical energy storage devices with form factors is launching a new era of competition. Herein, the first prototype planar sodium‐ion microcapacitors (NIMCs) are constructed based on the interdigital microelectrodes of urchin‐like sodium titanate as faradaic anode and nanoporous activated graphene as non‐faradaic cathode along with high‐voltage ionogel electrolyte on a single flexible substrate. By effectively coupling with battery‐type anode and capacitor‐type cathode, the resultant all‐solid‐state NIMCs working at 3.5 V exhibit a high volumetric energy density of 37.1 mWh cm−3 and an ultralow self‐discharge rate of 44 h from Vmax to 0.6 Vmax, both of which surpass most reported hybrid micro‐supercapacitors. Through tuning graphene layer covered on the top surface of interdigital microelectrodes, the NIMCs unveil remarkably enhanced power density, owing to the establishment of favorable multidirectional fast ion diffusion pathways that significantly reduce the charge transfer resistance. Meanwhile, the as‐fabricated NIMCs present excellent mechanical flexibility without capacitance fade under repeated deformation, and electrochemical stability at a high temperature of 80 °C because of using nonflammable ionogel electrolyte and in‐plane geometry. Therefore, these flexible planar NIMCs with multidirectional ion diffusion pathways hold tremendous potential for microelectronics. All‐solid‐state planar sodium‐ion microcapacitors are constructed on a single substrate based on faradaic sodium titanate as anode and non‐faradaic activated graphene as cathode along with nonflammable high‐voltage ionogel electrolyte. The resultant microcapacitors possess the merit of multidirectional fast ion diffusion pathways and show high volumetric performance, remarkable flexibility, and high‐temperature stability.
Journal Article
Get in shape : two-dimensional and three-dimensional shapes
\"Introduces the reader to two-dimensional and three-dimensional shapes.\"-- Provided by publisher.
Book
Mittel und Wege Elementarer Ebener Geometrie zur Quadratur des Kreises
Geometrische Figuren mit starkem Gestaltkontrast erschweren den Versuch, ihre Flächeninhaltsgrößen zu bestimmen und zu vergleichen. Das seit langem zitierte Beispiel QUADRATUR DES KREISES wird von der Mathematik als konstruktiv nicht lösbar beurteilt, bedingt durch die unvermeidliche Beteiligung der transzendenten Kreiszahl PI. Während die Mathematik sich auf den Kreis fokussiert, dessen PI-Abhängigkeit keinen Zweifel zulässt, konzentriert sich die Studie unter elementaren Bedingungen der Ebenen Geometrie auf das Quadrat. In einem übersichtlichen, konzentrischen, maßstäblich konstanten Arbeitsfeld ermittelt die STUDIE mit präzisen Konstruktionen und nachvollziehbaren Arbeitsschritten die Übereinstimmung der Flächeninhaltsgrößen von KREIS und QUADRAT. Sie weist nach, dass die RESTFLÄCHEN der ins Quadrat \"eingebetteten\" geometrischen Figuren INKREIS und konstruktiv BESTIMMTES INQUADRAT in ihren Flächeninhaltsgrößen übereinstimmen und damit die QUADRATUR DES KREISES beweisen.
eBook
The effectiveness of guided inquiry-based technology integration on pre-service mathematics teachers understanding of plane geometry
The study investigated the effects of a technology-integrated guided inquiry-based approach on pre-service mathematics teachers' conceptual understanding of geometry when compared to a guided inquiry approach and a traditional teacher-centered approach. A non-equivalent pretest-posttest control group quasi-experimental design was used. A three-stage sampling techniques was adopted. Two colleges were purposively selected and assigned to experimental and control groups through simple random sampling. A total of 116 pre-service primary mathematics teachers were assigned into three groups: experimental group1 (n = 48), experimental group2 (n = 38) and traditional group (n = 30). Pretest and posttest data were collected using a two-tiered test, and were analyzed with descriptive statistics, sample paired t-test and one-way ANOVA. Compared to the other two groups, pre-service mathematics teachers who received a technology-integrated guided inquiry-based approach showed a greater level of conceptual understanding. In accordance with the results, some recommendations were made for mathematics teacher educators.
Journal Article
Metric information in cognitive maps: Euclidean embedding of non-Euclidean environments
The structure of the internal representation of surrounding space, the so-called cognitive map, has long been debated. A Euclidean metric map is the most straight-forward hypothesis, but human navigation has been shown to systematically deviate from the Euclidean ground truth. Vector navigation based on non-metric models can better explain the observed behavior, but also discards useful geometric properties such as fast shortcut estimation and cue integration.
Journal Article
Rectangles and triangles : a song about drawing with shapes
Sing along while drawing an adventure with princesses, knights, and dragons, using rectangles and other shapes.
Book
Isoperimetric inequalities in unbounded convex bodies
We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body
eBook